Spreadsheet-based chi-squared test - Chi-squared distribution

Edexcel S3 2020 October Q2

9 marks Moderate -0.3

2. A university awards its graduates a degree in one of three categories, Distinction, Merit or Pass. Table 1 shows information about a random sample of 200 graduates from three departments, Arts, Humanities and Sciences. \begin{table}[h]

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	Arts	Humanities	Sciences	Total
Distinction	22	32	38	92
Merit	15	30	13	58
Pass	18	15	17	50
	Total	55	77	68

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

\end{table} Xiu wants to carry out a test of independence between the category of degree and the department. Table 2 shows some of the values of \(\frac { ( O - E ) ^ { 2 } } { E }\) for this test. \begin{table}[h]

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	Arts	Humanities	Sciences	Total
Distinction	0.43	0.33	1.44	2.20
Merit	0.06	2.63	2.29	4.98
Pass

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

\end{table}

Complete Table 2
Hence, complete Xiu’s hypothesis test using a \(5 \%\) level of significance. You should state the hypotheses, the degrees of freedom and the critical value used for this test.

OCR MEI Further Statistics A AS 2018 June Q5

13 marks Standard +0.3

5 A random sample of workers for a large company were asked whether they are smokers, ex-smokers or have never smoked. The responses were classified by the type of worker: Managerial, Production line or Administrative. Fig. 5 is a screenshot showing part of the spreadsheet used to analyse the data. Some values in the spreadsheet have been deliberately omitted. \begin{table}[h]

	A	B	C	D	E	F
1	Observed frequencies
2		Smoker	Ex-smoker	Never smoked	Totals
3	Managerial	2	10	5	17
4	Production line	18	15	21	54
5	Administrative	13	6	14	33
6	Totals	33	31	40	104
7
8		Expected frequencies
9		5.3942	5.0673	6.5385
10		17.1346		20.7692
11		10.4712	9.8365	12.6923
12
13		Contributions to the test statistic
14		2.1358	4.8017	0.3620
15		0.0437		0.0026
16			1.4964	0.1347
17				Test statistic	9.66
18

\captionsetup{labelformat=empty} \caption{Fig. 5}

\end{table}

(A) State the sample size.
(B) State the null and alternative hypotheses for a test to investigate whether there is any association between type of worker and smoking status.
Showing your calculations, find the missing values in each of the following cells.
- C 10
- C 15
- B 16
- Complete the hypothesis test at the \(10 \%\) level of significance.
- Discuss briefly what the data suggest about smoking status for different types of workers. You should make a comment for each type of worker.

OCR MEI Further Statistics A AS 2022 June Q5

14 marks Standard +0.3

5 A researcher is investigating whether there is any relationship between the overall performance of a student at GCSE and their grade in A Level Mathematics. Their A Level Mathematics grade is classified as A* or A, B, C or lower, and their overall performance at GCSE is classified as Low, Middle, High. Data are collected for a sample of 80 students in a particular area. The researcher carries out a chi-squared test. The screenshot below shows part of a spreadsheet used to analyse the data. Some values in the spreadsheet have been deliberately omitted.

1

A

B

C

D

E

\multirow{2}{*}{

}

Observed frequency

A* or A

B

C or lower

Totals

3

Low

6

13

9

28

4

Middle

10

6

8

24

5

High

15

10

3

28

6

Totals

31

29

20

80

7

8

\multirow{2}{*}{}

9

A* or A

B

C or lower

10

Low

10.85

11

Middle

9.30

12

High

10.85

13

\multirow[b]{2}{*}{Contribution to the test statistic}

14

15

A* or A

B

C or lower

16

Low

2.1680

0.8002

0.5714

17

Middle

0.0527

0.8379

0.6667

18

High

1.5873

2.2857

19

State what needs to be known about the sample for the test to be valid. For the remainder of this question, you should assume that the test is valid.
Determine the missing values in each of the following cells.
- C11
- C18
- In this question you must show detailed reasoning.
Carry out a hypothesis test at the \(10 \%\) significance level to investigate whether there is any association between level of performance at GCSE and A Level Mathematics grade.
Discuss briefly what the data suggest about A Level Mathematics grade for different levels of performance at GCSE.
State one disadvantage of using a 10\% significance level rather than a 5\% significance level in a hypothesis test.

OCR MEI Further Statistics A AS 2023 June Q6

15 marks Standard +0.3

6 An eight-sided dice has its faces numbered \(1,2 , \ldots , 8\).

In this part of the question you should assume that the dice is fair.
1. State the probability that, when the dice is rolled once, the score is at least 6 .
2. Show that the probability that the score is within 2 standard deviations of its mean is 1 .
A student thinks that the dice may be biased. To investigate this, the student decides to roll the dice 80 times and then carry out a \(\chi ^ { 2 }\) goodness of fit test of a uniform distribution. The spreadsheet below shows the data for the test, where some of the values have been deliberately omitted.
\multirow[b]{2}{*}{1} A B C D
Score Observed frequency Expected frequency Chi-squared contribution
2 1 14 10 1.6
3 2 4 10 3.6
4 3 10 10 0
5 4 15 10
6 5 6 10 1.6
7 6 11 10 0.1
8 7 7 10 0.9
9 8 10 0.9
1. Explain why all of the expected frequencies are equal to 10 .
2. Determine the missing values in each of the following cells.
  - B9
D5
(iii) In this question you must show detailed reasoning.

OCR MEI Further Statistics A AS 2020 November Q6

12 marks Standard +0.3

6 A researcher is investigating whether there is any relationship between whether a cyclist wears a helmet and the distance, \(x \mathrm {~m}\), the cyclist is from the kerb (the edge of the road). Data are collected at a particular location for a random sample of 250 cyclists. The researcher carries out a chi-squared test. Fig. 6 is a screenshot showing part of a spreadsheet used to analyse the data. Some values in the spreadsheet have been deliberately omitted. \begin{table}[h]

	A	B	C	D	E	F	G
1	\multirow{2}{*}{}		Observed frequency
2			\(\boldsymbol { x } \boldsymbol { \leq } \mathbf { 0 . 3 }\)	\(0.3 < x \leq 0.5\)	\(0.5 < x \leq 0.8\)	x > 0.8	Totals
3	\multirow[t]{2}{*}{Wears helmet}	Yes	26	27	23	46	122
4		No	45	31	21	31	128
5	\multirow{2}{*}{}	Totals	71	58	44	77	250
6
7			Expected frequency
8			\(\boldsymbol { x } \boldsymbol { \leq } \mathbf { 0 . 3 }\)	\(0.3 < x \leq 0.5\)	\(0.5 < x \leq 0.8\)	\(\boldsymbol { x } \boldsymbol { > } \mathbf { 0 . 8 }\)
9	\multirow[t]{2}{*}{Wears helmet}	Yes	34.6480			37.5760
10		No	36.3520			39.4240
11
12	\multirow{2}{*}{}		Contribution to the test statistic
13			\(\boldsymbol { x } \boldsymbol { \leq } \mathbf { 0 . 3 }\)	\(0.3 < x \leq 0.5\)	\(0.5 < x \leq 0.8\)	\(\boldsymbol { x } \boldsymbol { > } \mathbf { 0 . 8 }\)
14	\multirow[t]{2}{*}{Wears helmet}	Yes	2.1585	0.0601	0.1087	1.8885
15		No	2.0573	0.0573		1.8000
16

\captionsetup{labelformat=empty} \caption{Fig. 6}

\end{table}

Showing your calculations, find the missing values in each of the following cells.
- E10
- E15
- In this question you must show detailed reasoning.
Carry out a hypothesis test at the \(10 \%\) significance level to investigate whether there is any association between helmet wearing and distance from the kerb.
Discuss briefly what the data suggest about helmet wearing for different distances from the kerb.

OCR MEI Further Statistics Minor 2021 November Q3

13 marks Standard +0.3

3 A student wants to know whether there is any association between age and whether or not people smoke. The student takes a sample of 120 adults and asks each of them whether or not they smoke. Below is a screenshot showing part of a spreadsheet used to analyse the data. Some values in the spreadsheet have been deliberately omitted.

	A	B	C	D	E
1	\multirow{3}{*}{}		Observed frequency
2			Age
3			16-34	35-59	60 and over
4	\multirow{2}{*}{Smoking status}	Smoker	13	7	3
5		Non-smoker	28	43	26
6
7			Expected frequency
8			7.8583
9			33.1417
10
11			Contributions to the test statistic
12			3.3642	0.6964	1.1775
13				0.1651	0.2792
11

The student wants to carry out a chi-squared test to analyse the data. State a requirement of the sample if the test is to be valid. For the rest of this question, you should assume that this requirement is met.
Determine the missing values in each of the following cells.
- E8
- C13
- In this question you must show detailed reasoning.
Carry out a hypothesis test at the \(5 \%\) significance level to investigate whether there is any association between age and smoking status.
Discuss what the data suggest about the smoking status for each different age group.

OCR MEI Further Statistics Major 2019 June Q5

13 marks Standard +0.3

5 In an investigation into the possible relationship between smoking and weight in adults in a particular country, a researcher selected a random sample of 500 adults.
The adults in the sample were classified according to smoking status (non-smoker, light smoker or heavy smoker, where light smoker indicates less than 10 cigarettes per day) and body weight (underweight, normal weight or overweight). Fig. 5 is a screenshot showing part of the spreadsheet used to calculate the contributions for a chisquared test. Some values in the spreadsheet have been deliberately omitted. \begin{table}[h]

	A	B	C	D	E	F
1	Observed frequencies
2		Underweight	Normal	Overweight	Totals
3	Non-smoker	8	52	178	238
4	Light smoker	10	40	68	118
5	Heavy smoker	5	47	92	144
6	Totals	23	139	338	500
7
8	Expected frequencies
9	Non-smoker	10.9480	66.1640	160.8880
10	Light smoker	5.4280		79.7680
11	Heavy smoker		40.0320	97.3440
12
13
14	Non-smoker	0.7938		1.8200
15	Light smoker	3.8510	1.5785	1.7361
16	Heavy smoker	0.3982	1.2129	0.2934
17

\captionsetup{labelformat=empty} \caption{Fig. 5}

\end{table}

Showing your calculations, find the missing values in each of the following cells.
- B11
- C10
- C14
- Complete the hypothesis test at the \(1 \%\) level of significance.
- For each smoking status, give a brief interpretation of the largest of the three contributions to the test statistic.

OCR MEI Further Statistics Major Specimen Q9

13 marks Standard +0.3

9 A random sample of adults in the UK were asked to state their primary source of news: television (T), internet (I), newspapers (N) or radio (R). The responses were classified by age group, and an analysis was carried out to see if there is any association between age group and primary source of news. Fig. 9 is a screenshot showing part of the spreadsheet used to analyse the data. Some values in the spreadsheet have been deliberately omitted. \begin{table}[h]

	A	B	C	D	E	F
1	Source	Age group
2	of news	18-32	33-47	48-64	65+
3	T	63	61	71	80	275
4	I	33	33	22	12	100
5	N	9	8	11	20	48
6	R	4	9	9	5	27
7		109	111	113	117	450
8
9		Expected frequencies
10		66.61	67.83	69.06	71.50
11		24.22	24.67		26.00
12		11.63	11.84	12.05	12.48
13		6.54	6.66	6.78	7.02
14
15	Contributions to the test statistic
16		0.20	0.69	0.05	1.01
17		3.18	2.82		7.54
18		0.59		0.09	4.53
19		0.99	0.82	0.73	0.58
20	test statistic					25.45

\captionsetup{labelformat=empty} \caption{Fig. 9}

\end{table}

(A) State the sample size.
(B) Give the name of the appropriate hypothesis test.
(C) State the null and alternative hypotheses.
Showing your calculations, find the missing values in cells
- D11,
- D17 and
- C18.
- Complete the appropriate hypothesis test at the \(5 \%\) level of significance.
- Discuss briefly what the data suggest about primary source of news. You should make a comment for each age group.