Expected frequencies partially provided - Chi-squared test of independence

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18 marks Standard +0.3

4 A student is investigating whether there is any association between the species of shellfish that occur on a rocky shore and where they are located. A random sample of 160 shellfish is selected and the numbers of shellfish in each category are summarised in the table below.

		Location
\cline { 3 - 5 } \multicolumn{2}{\|c\|}{}	Exposed	Sheltered	Pool
\multirow{3}{*}{Species}	Limpet	24	32	16
\cline { 2 - 5 }	Mussel	24	11	3
\cline { 2 - 5 }	Other	5	22	23

Write down null and alternative hypotheses for a test to examine whether there is any association between species and location. The contributions to the test statistic for the usual $\chi ^ { 2 }$ test are shown in the table below.
Contribution Location
\cline { 3 - 5 } Exposed Sheltered Pool
\multirow{3}{*}{Species} Limpet 0.0009 0.2585 0.4450
\cline { 2 - 5 } Mussel 10.3472 1.2756 4.8773
\cline { 2 - 5 } Other 8.0719 0.1402 7.4298
The sum of these contributions is 32.85 .
Calculate the expected frequency for mussels in pools. Verify the corresponding contribution 4.8773 to the test statistic.
Carry out the test at the $5 \%$ level of significance, stating your conclusion clearly.
For each species, comment briefly on how its distribution compares with what would be expected if there were no association.
If 3 of the 160 shellfish are selected at random, one from each of the 3 types of location, find the probability that all 3 of them are limpets.

15 marks Standard +0.3

7 It is thought that a person's eye colour is related to the reaction of the person's skin to ultra-violet light. As part of a study, a random sample of 140 people were treated with a standard dose of ultra-violet light. The degree of reaction was classified as None, Mild or Strong. The results are given in Table 1. The corresponding expected frequencies for a $\chi ^ { 2 }$ test of association between eye colour and reaction are shown in Table 2. \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Table 1
Observed frequencies}

		Eye colour
		Blue	Brown	Other	Total
	None	12	17	10	39
Reaction	Mild	31	21	11	63
	Strong	22	4	12	38
	Total	65	42	33	140

\end{table} \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Table 2
Expected frequencies}

		Eye colour
		Blue	Brown	Other
	None	18.11	11.70	9.19
Reaction	Mild	29.25	18.90	14.85
	Strong	17.64	11.40	8.96

\end{table}

(a) State suitable hypotheses for the test.
(b) Show how the expected frequency of 18.11 in Table 2 is obtained.
(c) Show that the three cells in the top row together contribute 4.53 to the calculated value of $\chi ^ { 2 }$, correct to 2 decimal places.
(d) You are given that the total calculated value of $\chi ^ { 2 }$ is 12.78 , correct to 2 decimal places. Give the smallest value of $\alpha$ obtained from the tables for which the null hypothesis would be rejected at the $\alpha \%$ significance level.
Test, at the $5 \%$ significance level, whether the proportions of people in the whole population with blue eyes, brown eyes and other colours are in the ratios $2 : 2 : 1$.

15 marks Standard +0.3

6 The Research and Development department of a paint manufacturer has produced paint of three different shades of grey, $G _ { 1 } , G _ { 2 }$ and $G _ { 3 }$. In order to find the reaction of the public to these shades, each of a random sample of 120 people was asked to state which shade they preferred. The results, classified by gender, are shown in Table 1. \begin{table}[h]

Shade
\cline { 2 - 5 }		$G _ { 1 }$	$G _ { 2 }$	$G _ { 3 }$
\cline { 2 - 5 } Gender	Male	11	24	23
Female	18	13	31
\cline { 2 - 5 }
\cline { 2 - 5 }

\captionsetup{labelformat=empty} \caption{Table 1}

\end{table} Table 2 shows the corresponding expected values, correct to 2 decimal places, for a test of independence. \begin{table}[h]

Shade
\cline { 2 - 5 }		$G _ { 1 }$	$G _ { 2 }$	$G _ { 3 }$
\cline { 2 - 5 } Gender	Male	14.02	17.88	26.10
	Female	14.98	19.12	27.90
\cline { 2 - 5 }
\cline { 2 - 5 }

\captionsetup{labelformat=empty} \caption{Table 2}

\end{table}

Show how the value 17.88 for Male, $G _ { 2 }$ was obtained.
Test, at the $5 \%$ significance level, whether gender and preferred shade are independent.
Determine the smallest significance level obtained from tables or calculator for which there is evidence that not all shades are equally preferred by people in general, irrespective of gender.

9 marks Moderate -0.3

5 Gloria is a market trader who sells jeans. She trades on Mondays, Wednesdays and Fridays. Wishing to investigate whether the volume of trade depends on the day of the week, Gloria analysed a random sample of 150 days' sales and classified them by day and volume (low, medium and high). The results are given in the table below.

		Day
		Monday	Wednesday	Friday
\multirow{3}{*}{Volume}	Low	15	13	2
	Medium	23	26	23
	High	12	9	27

Gloria asked a statistician to perform a suitable test of independence and, as part of this test, expected frequencies were calculated. These are shown in the table below.

		Day
		Monday	Wednesday	Friday
	Low	10.00	9.60	10.40
Volume	Medium	24.00	23.04	24.96
	High	16.00	15.36	16.64

Show how the value 23.04 for medium volume on Wednesday has been obtained.
State, giving a reason, if it is necessary to combine any rows or columns in order to carry out the test. The value of the test statistic is found to be 21.15, correct to 2 decimal places.
Stating suitable hypotheses for the test, give its conclusion using a $1 \%$ significance level. Gloria wishes to hold a sale and asks the statistician to advise her on which day to hold it in order to sell as much as possible.
State the day that the statistician should advise and give a reason for the choice.

18 marks Moderate -0.3

4 A council provides waste paper recycling services for local businesses. Some businesses use the standard service for recycling paper, others use a special service for dealing with confidential documents, and others use both. Businesses are classified as small or large. A survey of a random sample of 285 businesses gives the following data for size of business and recycling service.

Recycling Service

\cline { 3 - 5 } \multicolumn{2}{|c|}{}

Standard

Special

Both

Size of

business

Small

35

26

44

Large

55

52

73

Write down null and alternative hypotheses for a test to examine whether there is any association between size of business and recycling service used. The contributions to the test statistic for the usual $\chi ^ { 2 }$ test are shown in the table below.
Recycling Service
\cline { 3 - 5 } \multicolumn{2}{|c|}{} Standard Special Both
Size of
business
Small 0.1023 0.2607 0.0186
Large 0.0597 0.1520 0.0108
The sum of these contributions is 0.6041 .
Calculate the expected frequency for large businesses using the special service. Verify the corresponding contribution 0.1520 to the test statistic.
Carry out the test at the $5 \%$ level of significance, stating your conclusion clearly. The council is also investigating the weight of rubbish in domestic dustbins. In 2008 the average weight of rubbish in bins was 32.8 kg . The council has now started a recycling initiative and wishes to determine whether there has been a reduction in the weight of rubbish in bins. A random sample of 50 domestic dustbins is selected and it is found that the mean weight of rubbish per bin is now 30.9 kg , and the standard deviation is 3.4 kg .
Carry out a test at the $5 \%$ level to investigate whether the mean weight of rubbish has been reduced in comparison with 2008 . State carefully your null and alternative hypotheses. www.ocr.org.uk after the live examination series.
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17 marks Moderate -0.3

4 Birds are observed at feeding stations in three different places - woodland, farm and garden. The numbers of finches, thrushes and tits observed at each site are summarised in the table. The birds observed are regarded as a random sample from the population of birds of these species that use these feeding stations.

\multirow{2}{*}{Observed Frequency}		Place
		Farm	Garden	Woodland	Totals
\multirow{4}{*}{Species}	Thrushes	11	74	7	92
	Tits	70	26	88	184
	Finches	17	2	10	29
	Totals	98	102	105	305

The expected frequencies under the null hypothesis for the usual $\chi ^ { 2 }$ test are shown in the table below.

\multirow{2}{*}{Expected Frequency}		Place
		Farm	Garden	Woodland
\multirow{3}{*}{Species}	Thrushes	29.5607	30.7672	31.6721
	Tits	59.1213	61.5344	63.3443
	Finches	9.3180	9.6984	9.9836

Verify that the entry 9.3180 is correct. The corresponding contributions to the test statistic are shown in the table below.
\multirow{2}{*}{Contribution} Place
Farm Garden Woodland
\multirow{3}{*}{Species} Thrushes 11.6539 60.7489 19.2192
Tits 2.0017 20.5201 9.5969
Finches 6.3332 6.1108 0.0000
Verify that the entry 6.3332 is correct.
Carry out the test at the $1 \%$ level of significance.
For each place, use the table of contributions to comment briefly on the differences between the observed and expected distributions of species.

18 marks Standard +0.3

4

In a survey on internet usage, a random sample of 200 people is selected. The people are asked how much they have spent on internet shopping during the last three months. The results, classified by amount spent and sex, are shown in the table.

\multirow{2}{*}{}		Sex		\multirow{2}{*}{Row totals}
		Male	Female
\multirow{5}{*}{Amount spent}	Nothing	28	34	62
	Less than £50	17	21	38
	£50 up to £200	22	26	48
	£200 up to £1000	23	16	39
	£1000 or more	8	5	13
Column totals		98	102	200

Write down null and alternative hypotheses for a test to examine whether there is any association between amount spent and sex of person. The contributions to the test statistic for the usual $\chi ^ { 2 }$ test are shown in the table below.
\multirow{2}{*}{} Sex
Male Female
\multirow{5}{*}{Amount spent} Nothing 0.1865 0.1791
Less than £50 0.1409 0.1354
£50 up to £200 0.0982 0.0944
£200 up to £1000 0.7918 0.7608
£1000 or more 0.4171 0.4007
The sum of these contributions, correct to 3 decimal places, is 3.205.
Calculate the expected frequency for females spending nothing. Verify the corresponding contribution, 0.1791 , to the test statistic.
Carry out the test at the $5 \%$ level of significance, stating your conclusion clearly.

A bakery sells loaves specified as having a mean weight of 400 grams. It is known that the weights of these loaves are Normally distributed and that the standard deviation is 5.7 grams. An inspector suspects that the true mean weight may be less than 400 grams. In order to test this, the inspector takes a random sample of 6 loaves. Carry out a suitable test at the $5 \%$ level, given that the weights, in grams, of the 6 loaves are as follows. $\begin{array} { l l l l l l } 392.1 & 405.8 & 401.3 & 387.4 & 391.8 & 400.6 \end{array}$ RECOGNISING ACHIEVEMENT

17 marks Standard +0.3

4

Mary is opening a cake shop. As part of her market research, she carries out a survey into which type of cake people like best. She offers people 4 types of cake to taste: chocolate, carrot, lemon and ginger. She selects a random sample of 150 people and she classifies the people as children and adults. The results are as follows.

\multirow{2}{*}{}		Classification of person		\multirow{2}{*}{Row totals}
		Child	Adult
\multirow{4}{*}{Type of cake}	Chocolate	34	23	57
	Carrot	16	18	34
	Lemon	4	18	22
	Ginger	13	24	37
Column totals		67	83	150

The contributions to the test statistic for the usual $\chi ^ { 2 }$ test are shown in the table below.

Classification of person

\cline { 3 - 4 } \multicolumn{2}{|c|}{}

Child

Adult

\multirow{3}{*}{

Type

of

cake

}

Chocolate

2.8646

2.3124

\cline { 2 - 4 }

Carrot

0.0436

0.0352

\cline { 2 - 4 }

Lemon

3.4549

2.7889

\cline { 2 - 4 }

Ginger

0.7526

0.6075

The sum of these contributions, correct to 2 decimal places, is 12.86 .

Calculate the expected frequency for children preferring chocolate cake. Verify the corresponding contribution, 2.8646, to the test statistic.
Carry out the test at the $1 \%$ level of significance.

Mary buys flour in bags which are labelled as containing 5 kg . She suspects that the average contents of these bags may be less than 5 kg . In order to test this, she selects a random sample of 8 bags and weighs their contents. Assuming that weights are Normally distributed with standard deviation 0.0072 kg , carry out a test at the $5 \%$ level, given that the weights of the 8 bags in kg are as follows.
4.992
4.981
4.982
4.996
4.991
5.006
5.009
5.003
[0pt] [9] }{www.ocr.org.uk}) after the live examination series.
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OCR is part of the

20 marks Moderate -0.3

4

A random sample of 80 GCSE students was selected to take part in an investigation into whether attitudes to mathematics differ between girls and boys. The students were asked if they agreed with the statement 'Mathematics is one of my favourite subjects'. They were given three options 'Agree', 'Disagree', 'Neither agree nor disagree'. The results, classified according to sex, are summarised in the table below.
Agree Disagree Neither
Male 17 13 8
Female 12 11 19
The contributions to the test statistic for the usual $\chi ^ { 2 }$ test are shown in the table below.
Agree Disagree Neither
Male 0.7550 0.2246 1.8153
Female 0.6831 0.2032 1.6424
1. Calculate the expected frequency for females who agree. Verify the corresponding contribution, 0.6831 , to the test statistic.
2. Carry out the test at the $5 \%$ level of significance.
The level of radioactivity in limpets (a type of shellfish) in the sea near to a nuclear power station is regularly monitored. Over a period of years it has been found that the level (measured in suitable units) is Normally distributed with mean 5.64. Following an incident at the power station, a researcher suspects that the mean level of radioactivity in limpets may have increased. The researcher selects a random sample of 60 limpets. Their levels of radioactivity, $x$ (measured in the same units), are summarised as follows. $$\sum x = 373 \quad \sum x ^ { 2 } = 2498$$ Carry out a test at the $5 \%$ significance level to investigate the researcher's belief.

9 marks Standard +0.3

7 A dairy industry researcher, Robyn, decided to investigate the milk yield, classified as low, medium or high, obtained from four different breeds of cow, $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D . The milk yield of a sample of 105 cows was monitored and the results are summarised in contingency Table 1.

\multirow{2}{*}{Table 1}		Yield
		Low	Medium	High	Total
\multirow{4}{*}{Breed}	A	4	5	12	21
	B	10	6	4	20
	C	8	17	7	32
	D	5	20	7	32
	Total	27	48	30	105

The sample of cows may be regarded as random.
Robyn decides to carry out a $\chi ^ { 2 }$-test for association between milk yield and breed using the information given in Table 1. 7

Contingency Table 2 gives some of the expected frequencies for this test.
Complete Table 2 with the missing expected values.
\multirow[t]{2}{*}{Table 2} Yield
Low Medium High
\multirow{4}{*}{Breed} A 6
B 5.14 9.14 5.71
C
D 8.23 14.63 9.14
7
(i) For Robyn's test, the test statistic $\sum \frac { ( O - E ) ^ { 2 } } { E } = 19.4$ correct to three significant figures.
Use this information to carry out Robyn's test, using the $1 \%$ level of significance.
7 (b) (ii) By considering the observed frequencies given in Table 1 with the expected frequencies in Table 2, interpret, in context, the association, if any, between milk yield and breed.

8 marks Standard +0.3

2 For a random sample of 160 employees of a large company, the principal method of transport for getting to work, arranged according to grade of employee, is shown in the table.

Grade	Walk or cycle	Private motorised transport	Public transport
A	9	13	6
B	16	43	41
C	11	8	13

A test is carried out at the $5 \%$ significance level of whether there is association between grade of employee and method of transport.

State appropriate hypotheses for the test. The contributions to the test statistic are shown in the following table, correct to 3 decimal places.
Grade Walk or cycle Private motorised transport Public transport
A 1.157 0.289 1.929
B 1.878 0.225 0.327
C 2.006 1.800 0.083
Show how the value 0.225 is obtained.
Complete the test, stating the conclusion.
Which combination of grade of employee and method of transport most strongly suggests association? Justify your answer.

12 marks Standard +0.3

John thinks that a person's eye colour is related to their hair colour. He takes a random sample of 600 people and records their eye and hair colours. The results are shown in Table 1.

\begin{table}[h]

\multirow{2}{*}{}		Hair colour
		Black	Brown	Red	Blonde	Total
\multirow{5}{*}{Eye colour}	Brown	45	125	15	58	243
	Blue	34	90	10	58	192
	Hazel	20	38	16	26	100
	Green	6	29	7	23	65
	Total	105	282	48	165	600

\captionsetup{labelformat=empty} \caption{Table 1}

\end{table} John carries out a $\chi ^ { 2 }$ test in order to test whether eye colour and hair colour are related. He calculates the expected frequencies shown in Table 2. \begin{table}[h]

\multirow{2}{*}{}		Hair colour
		Black	Brown	Red	Blonde
\multirow{4}{*}{Eye colour}	Brown	42.5	114.2	19.4	66.8
	Blue	33.6	90.2	15.4	52.8
	Hazel	17.5	47	8	27.5
	Green	11.4	30.6	5.2	17.9

\captionsetup{labelformat=empty} \caption{Table 2}

\end{table}

Show how the value 47 in Table 2 has been calculated.
Write down the number of degrees of freedom John should use in this $\chi ^ { 2 }$ test. Given that the value of the $\chi ^ { 2 }$ statistic is 20.6 , to 3 significant figures,
find the smallest value of $\alpha$ for which the null hypothesis will be rejected at the $\alpha \%$ level of significance.
Use the data from Table 1 to test at the $5 \%$ level of significance whether or not the proportions of people in the population with black, brown, red and blonde hair are in the ratio 2:6:1:3 State your hypotheses clearly.

7 marks Standard +0.3

A survey asked a random sample of 200 people their age and the main use of their mobile phone.

The results are shown in Table 1 below. \begin{table}[h]

\multirow{2}{*}{}		Main use of their mobile phone
		Internet	Texts	Phone calls
\multirow{3}{*}{Age}	Under 20	27	14	9
	From 20 to 40	32	34	29
	Over 40	15	19	21

\captionsetup{labelformat=empty} \caption{Table 1}

\end{table} The data are to be used to test whether or not age and main use of their mobile phone are independent. Table 2 shows the expected frequencies for each group, assuming people's age and main use of their mobile phone are independent. \begin{table}[h]

\multirow{2}{*}{}		Main use of their mobile phone
		Internet	Texts	Phone calls
\multirow{3}{*}{Age}	Under 20	18.5	16.75	14.75
	From 20 to 40	35.15	31.825	28.025
	Over 40	20.35	18.425	16.225

\captionsetup{labelformat=empty} \caption{Table 2}

\end{table}

For users under 20 choosing the Internet as the main use of their mobile phone,
1. verify that the expected frequency is 18.5
2. show that the contribution to the $\chi ^ { 2 }$ test statistic is 3.91 to 3 significant figures.
Given that the $\chi ^ { 2 }$ test statistic for the data is 9.893 to 3 decimal places, test at the $5 \%$ level of significance whether or not age and main use of their mobile phone are independent. State your hypotheses clearly.

14 marks Standard +0.3

4. A psychologist carries out a survey of the perceived body weight of 150 randomly chosen people. He asks them if they think they are underweight, about right or overweight. His results are summarised in the table below.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Underweight	About right	Overweight
Male	20	22	30
Female	16	28	34

The psychologist calculates two of the expected frequencies, to 2 decimal places, for a test of independence between perceived body weight and gender. These results are shown in the table below.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Underweight	About right	Overweight
Male	17.28
Female	18.72

Complete the table of expected frequencies shown above.
Test, at the $10 \%$ level of significance, whether or not perceived body weight is independent of gender. State your hypotheses clearly. The psychologist now combines the male and female data to test whether or not body weight types are chosen equally.
Find the smallest significance level, from the tables in the formula booklet, for which there is evidence of a preference.

13 marks Standard +0.3

10 A scientist is researching dietary fat intake and cholesterol level. A random sample of 60 people is selected and their dietary fat intakes and cholesterol levels are measured. Dietary fat intakes are classified as low, medium and high, and cholesterol levels are classified as normal and high. The scientist decides to carry out a chi-squared test to investigate whether there is any association between dietary fat intake and cholesterol level. Tables $\mathbf { 1 0 . 1 }$ and $\mathbf { 1 0 . 2 }$ show the data and some of the expected frequencies for the test. \begin{table}[h]

\multirow{2}{*}{}		Dietary fat intake
		Low	Medium	High	Total
\multirow{2}{*}{Cholesterol level}	Normal	9	18	5	32
	High	3	13	12	28
	Total	12	31	17	60

\captionsetup{labelformat=empty} \caption{Table 10.1}

\end{table} \begin{table}[h]

Expected frequency

Dietary fat intake

\cline { 3 - 5 }

Low

Medium

High

\multirow{2}{*}{

Cholesterol

level

}

Normal

9.0667

\cline { 2 - 5 }

High

7.9333

\captionsetup{labelformat=empty} \caption{Table 10.2}

\end{table}

Complete the table of expected frequencies in the Printed Answer Booklet.

Determine the contribution to the chi-squared test statistic for people with normal cholesterol level and high dietary fat intake, giving your answer to $\mathbf { 4 }$ decimal places. The contributions to the chi-squared test statistic for the remaining categories are shown in Table 10.3. \begin{table}[h]

Dietary fat intake

\cline { 2 - 5 }

Low

Medium

High

\multirow{2}{*}{

Cholesterol

level

}

Normal

1.0563

0.1301

\cline { 2 - 5 }

High

1.2071

0.1487

2.0846

\captionsetup{labelformat=empty} \caption{Table 10.3} \end{table}

In this question you must show detailed reasoning. Carry out the test at the 5\% significance level.

For each level of dietary fat intake, give a brief interpretation of what the data suggest about the level of cholesterol.

13 marks Standard +0.3

9 A cyclist has 3 bicycles, a road bike, a gravel bike and an electric bike. She wishes to know if the bicycle which she is riding makes any difference to whether she reaches a speed of 25 mph or greater on a journey. She selects a random sample of 120 journeys and notes the bicycle and whether or not her maximum speed was 25 mph or greater. She decides to carry out a chisquared test to investigate whether there is any association between bicycle type and whether her maximum speed is 25 mph or greater. Tables 9.1 and 9.2 show the data and some of the expected frequencies for the test. \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Table 9.1}

\multirow{2}{*}{}		Bicycle
		Road	Gravel	Electric	Total
\multirow{2}{*}{Maximum speed}	Less than 25 mph	2	21	19	42
	25 mph or greater	13	47	18	78
	Total	15	68	37	120

\end{table} \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Table 9.2}

\multirow{2}{*}{Expected frequency}		Bicycle
		Road	Gravel	Electric
\multirow{2}{*}{Maximum speed}	Less than 25 mph			12.95
	25 mph or greater			24.05

\end{table}

Complete the table of expected frequencies in the Printed Answer Booklet.
Determine the contribution to the chi-squared test statistic for the Electric bicycle and maximum speed 25 mph or greater. Give your answer correct to 4 decimal places. The contributions to the chi-squared test statistic for the remaining categories are shown in Table 9.3. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Table 9.3}
\multirow{2}{*}{Contribution to the test statistic} Bicycle
Road Gravel Electric
\multirow{2}{*}{Maximum speed} Less than 25 mph 2.0119 0.3294 2.8264
25 mph or greater 1.0833 0.1774
\end{table}
In this question you must show detailed reasoning. Carry out the test at the 5\% significance level.
For each type of bicycle, give a brief interpretation of what the data suggest about maximum speed.

11 marks Standard +0.3

6. An online survey on the use of social media asked the following question: \begin{displayquote} "Do you use any form of social media?" \end{displayquote} The results for a total of 1953 respondents are shown in the table below.

	Age in years
Use social media	18-29	30-49	50-64	65 or older	Total
Yes	310	412	348	196	1266
No	42	116	196	333	687
Total	352	528	544	529	1953

To test whether there is a relationship between social media use and age, a significance test is carried out at the $5 \%$ level.

State the null and alternative hypotheses.
Show how the expected frequency $228 \cdot 18$ is calculated in the table below.
Expected values Age in years
Use social media 18-29 30-49 50-64 65 or older
Yes $228 \cdot 18$ $342 \cdot 27$ 352.64 342.92
No 123.82 185.73 191.36 186.08
Determine the value of $s$ in the table below.
Chi-squared contributions Age in years
Use social media 18-29 30-49 50-64 65 or older
Yes 29.34 $s$ 0.06 62.94
No 54.07 26-18 0.11 115.99
Complete the significance test, showing all your working.
A student, analysing these data on a spreadsheet, obtains the following output. \includegraphics[max width=\textwidth, alt={}, center]{77fd7ad7-f5a3-4947-afc6-e5ef45bef7a8-5_202_1271_445_415} Explain why the student must have made an error in calculating the $p$-value.

7 marks Standard +0.3

Abram carried out a survey of two treatments for a plant fungus. The contingency table below shows the results of a survey of a random sample of 125 plants with the fungus.

\multirow{2}{*}{}		Treatment
		No action	Plant sprayed once	Plant sprayed every day
\multirow{3}{*}{Outcome}	Plant died within a month	15	16	25
	Plant survived for 1-6 months	8	25	10
	Plant survived beyond 6 months	7	14	5

Abram calculates expected frequencies to carry out a suitable test. Seven of these are given in the partly-completed table below.

\multirow{2}{*}{}		Treatment
		No action	Plant sprayed once	Plant sprayed every day
\multirow{3}{*}{Outcome}	Plant died within a month			17.92
	Plant survived for 1-6 months	10.32	18.92	13.76
	Plant survived beyond 6 months	6.24	11.44	8.32

The value of $\sum \frac { ( O - E ) ^ { 2 } } { E }$ for the 7 given values is 8.29
Test at the $2.5 \%$ level of significance, whether or not there is an association between the treatment of the plants and their survival. State your hypotheses and conclusion clearly.

6 marks Standard +0.3

Tisam took a survey of students' favourite colours. The results are summarised in the table below.

\multirow{2}{*}{}		Colour
		Red	Blue	Green	Yellow	Black	Total
\multirow{3}{*}{Year group}	1-5	34	15	14	22	3	88
	6-9	23	32	12	9	8	84
	10-12	5	28	19	8	8	68
	Total	62	75	45	39	19	240

Tisam carries out a suitable test to see if there is any association between favourite colour and year group.

Write down the hypotheses for a suitable test. For her table, Tisam only needs to check one cell to show that none of the expected frequencies are less than 5
1. Identify this cell, giving your reason.
2. Calculate the expected frequency for this cell. The test statistic for Tisam's test is 38.449
Using a $1 \%$ level of significance, complete the test. You should state your critical value and conclusion clearly.

10 marks Standard +0.3

8 An insurance company groups its vehicle insurance policies into two categories, car insurance and motorbike insurance. The number of claims in a random sample of 80 policies was monitored and the results summarised in contingency Table 1. \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Table 1}

\end{table} The insurance company decides to carry out a $\chi ^ { 2 }$-test for association between number of claims and type of insurance policy using the information given in Table 1. 8

The contingency table shown in Table 2 gives some of the exact expected frequencies for this test. Complete Table 2 with the missing exact expected values. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Table 2}
\multirow{2}{*}{} Number of claims
0 1 2 3 or more
\multirow{2}{*}{Type of insurance policy} Car 10.0625 4.375
Motorbike 10.6875
\end{table} 8
Carry out the insurance company's test, using the $10 \%$ level of significance. \includegraphics[max width=\textwidth, alt={}, center]{313cd5ce-07ff-4781-a134-565b8b221145-12_2488_1719_219_150} Additional page, if required.
Write the question numbers in the left-hand margin. Additional page, if required.
Write the question numbers in the left-hand margin. Additional page, if required.
Write the question numbers in the left-hand margin.

8 marks Standard +0.3

7 Wade and Odelia are investigating whether there is an association between the region where a person lives and the brand of washing powder they use. They decide to conduct a $\chi ^ { 2 }$-test for association and survey a random sample of 200 people. The expected frequencies for the test have been calculated and are shown in the contingency table below.

\multirow{2}{*}{}

Number of claims

\multirow[b]{3}{*}{Type of insurance policy}

10 marks Moderate -0.3

A student, considering options for the future, collects data on education and salary. The table below shows the highest level of education attained and the salary bracket of a random sample of 664 people.

	Fewer than 5 GCSE	5 or more GCSE	3 A Levels	University degree	Post graduate qualification	Total
Less than £20 000	18	32	20	28	10	108
£20 000 to £60 000	50	95	112	155	50	462
More than £60 000	3	22	29	35	5	94
Total	71	149	161	218	65	664

By conducting a chi-squared test for independence, the student investigates the relationship between the highest level of education attained and the salary earned.

State the null and alternative hypotheses. [1]

The table below shows the expected values. Calculate the value of $k$. [2]

Expected values	Fewer than 5 GCSE	5 or more GCSE	3 A Levels	University degree	Post graduate qualification
Less than £20 000	$k$	24·23	26·19	35·46	10·57
£20 000 to £60 000	49·40	103·67	112·02	151·68	45·23
More than £60 000	10·05	21·09	22·79	30·86	9·20

The following computer output is obtained. Calculate the values of $m$ and $n$. [2]

Chi Squared Contributions	Fewer than 5 GCSE	5 or more GCSE	3 A Levels	University degree	Post graduate qualification
Less than £20 000	3·604530799	$m$	1·46165	1·5686	0·03098
£20 000 to £60 000	0·007272735	0·72535	4E-06	0·07264	0·50396
More than £60 000	4·946619863	0·03897	1·69081	0·55498	$n$

X-squared = 19·61301, df = 8, p-value = 0·0119

1. Without carrying out any further calculations, explain how X-squared = 19·61301 (the $\chi^2$ test statistic) was calculated. [2]
2. Comment on the values in the "Fewer than 5 GCSE" column of the table in part (c). [2]
The student says that the highest levels of education lead to the highest paying jobs. Comment on the accuracy of the student's statement. [1]

Contribution		Location
\cline { 3 - 5 }	Exposed	Sheltered	Pool
\multirow{3}{*}{Species}	Limpet	0.0009	0.2585	0.4450
\cline { 2 - 5 }	Mussel	10.3472	1.2756	4.8773
\cline { 2 - 5 }	Other	8.0719	0.1402	7.4298

Shade
\cline { 2 - 5 }		\(G _ { 1 }\)	\(G _ { 2 }\)	\(G _ { 3 }\)
\cline { 2 - 5 } Gender	Male	11	24	23
Female	18	13	31
\cline { 2 - 5 }
\cline { 2 - 5 }

\multirow{2}{*}{Contribution}		Place
		Farm	Garden	Woodland
\multirow{3}{*}{Species}	Thrushes	11.6539	60.7489	19.2192
	Tits	2.0017	20.5201	9.5969
	Finches	6.3332	6.1108	0.0000

	Agree	Disagree	Neither
Male	0.7550	0.2246	1.8153
Female	0.6831	0.2032	1.6424

\multirow[t]{2}{*}{Table 2}		Yield
		Low	Medium	High
\multirow{4}{*}{Breed}	A			6
	B	5.14	9.14	5.71
	C
	D	8.23	14.63	9.14

Grade	Walk or cycle	Private motorised transport	Public transport
A	1.157	0.289	1.929
B	1.878	0.225	0.327
C	2.006	1.800	0.083