5.06b Fit prescribed distribution: chi-squared test

6. \begin{figure}[h] \end{figure} The sketch in Figure 1 represents a target which consists of 4 regions formed from 4 concentric circles of radii \(4 \mathrm {~cm} , 7 \mathrm {~cm} , 9 \mathrm {~cm}\) and 10 cm . The regions are coloured as labelled in Figure 1.
A random sample of 100 children each choose a point on the target and their results are summarised in the table below. (b) Find the value of \(r\) and the value of \(s\). Henry obtained a test statistic of 6.188 and no groups were pooled.
(c) State what conclusion Henry should make about his claim. Phoebe believes that the children chose the region of the target according to colour. She believes that boys and girls would favour different colours and splits the original data by gender to obtain the following table. \section*{Observed frequencies} (d) State suitable hypotheses to test Phoebe's belief. Phoebe calculated the following expected frequencies to carry out a suitable test. \section*{Expected frequencies} (e) Show how the value of 25.35 was obtained. Phoebe carried out the test using 2 degrees of freedom and a \(10 \%\) level of significance. She obtained a test statistic of 1.411
(f) Explain clearly why Phoebe used 2 degrees of freedom.
(g) Stating your critical value clearly, determine whether or not these data support Phoebe's belief.

2. \begin{figure}[h] \end{figure} The pointer shown in Figure 1 is spun so that it comes to rest between 0 and 360 degrees.
Linda claims that it is equally likely to come to rest at any point between 0 and 360 degrees. She spins the pointer 100 times and her results are summarised in the table below. She calculates expected frequencies for some of the possible outcomes and these are also given in the table below.

1. Find the values of the missing expected frequencies \(a , b\) and \(c\).
2. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test whether or not Linda's claim is supported by these data.

1. David carries out an experiment with 4 identical dice, each with faces numbered 1 to 6 . He rolls the 4 dice and counts the number of dice showing an even number on the uppermost face. He repeats this 150 times. The results are summarised in the table below.

David defines the random variable \(C\) as the number of dice showing an even number on the uppermost face when the four dice are thrown. David claims that \(C \sim \mathrm {~B} ( 4,0.5 )\)

1. Stating your hypotheses clearly and using a \(1 \%\) level of significance, test David's claim. Show your working clearly. John claims that \(C \sim \mathrm {~B} ( 4 , p )\)
2. Calculate an estimate of the value of \(p\) from the summary of the results of David's experiment. Show your working clearly. John decides to test his claim. He calculates expected frequencies using the results of David's experiment and obtains the following table.

   No. of dice showing an even number	0	1	2	3	4
   Expected frequency	8.65	36.00	\(d\)	39.00	\(e\)

3. Calculate, to 2 decimal places, the value of \(d\) and the value of \(e\)
4. State suitable hypotheses to test John's claim. John obtained a test statistic of 16.9 and carries out a test at the \(1 \%\) level of significance.
5. State what conclusion John should make about his claim.

   END

7. A student collects data on whether competitors in local tennis tournaments are right, or left-handed. The table below shows the number of left-handed players who reached the last 16 for fifty tournaments. The student believes that a binomial distribution with \(n = 16\) and \(p = 0.1\) could be a suitable model for these data.

1. Stating your hypotheses clearly test the student's model at the \(5 \%\) level of significance.
   (13 marks)
   To improve the model the student decides to estimate \(p\) using the data in the table. Using this value of \(p\) to calculate expected frequencies the student had 5 classes after combining and calculated that \(\sum \frac { ( O - E ) ^ { 2 } } { E } = 2.127\)
2. Test at the \(5 \%\) level of significance whether or not the binomial distribution is a suitable model for the number of left-handed players who reach the last 16 in local tennis tournaments. \section*{END}

1. A psychologist is investigating the numbers people choose when asked to pick a number at random in a given interval. He finds that when asked to pick a number between 0 and 100 people are less likely to pick certain numbers, such as multiples of ten. He believes, however that if people are asked to pick an odd number between 0 and 100 they are equally likely to pick a number ending in any of the digits \(1,3,5,7\) or 9 .

To test this theory he asks 80 people to pick an odd number between 0 and 100 and records the last digit of the numbers chosen. His results are shown in the table below. Stating your hypotheses clearly and using a 10\% level of significance test the psychologist's theory.
(9 marks)

4. A paranormal investigator invites couples who believe they have a telepathic connection to participate in a trial. With each couple one person looks at a card with one of five shapes on it and the other person says which of the shapes they think it is. This is repeated six times and the number of correct answers recorded. The results from 120 couples are given below. The investigator wishes to see if this data fits a binomial distribution with parameters \(n = 6\) and \(p = \frac { 1 } { 5 }\) and calculates to 2 decimal places the expected frequencies given below.

1. Find the other expected frequencies.
2. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not the distribution is an appropriate model.
3. Comment on your findings.

6 An eight-sided dice has its faces numbered \(1,2 , \ldots , 8\). \begin{enumerate}[label=(\alph*)] \item 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 .

\item 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.

1. Explain why all of the expected frequencies are equal to 10 .
2. Determine the missing values in each of the following cells.
   Carry out the \(\chi ^ { 2 }\) test at the \(5 \%\) significance level.

3 A glassware factory produces a large number of ornaments each week. Just before they leave the factory, all the ornaments are checked and some may be found to be defective. The Quality Assurance Manager of the factory wishes to model the number of defective ornaments that are found each week using a Poisson distribution. The numbers of defective ornaments found each week in a period of 40 weeks are shown in Table 3.1. \begin{table}[h] \end{table} You are given that summary statistics for the data are \(\sum f = 40 , \sum \mathrm { rf } = 84\) and \(\sum \mathrm { r } ^ { 2 } \mathrm { f } = 256\).

1. By using the summary statistics to determine estimates for the mean and variance of the number of defective ornaments produced by the factory each week, explain how the data support the suggestion that the number of defective ornaments produced each week can be modelled using a Poisson distribution. The Quality Assurance Manager is asked by the head office to carry out a chi-squared hypothesis test for goodness of fit based on a \(\operatorname { Po } ( 2 )\) distribution.
2. Table 3.2, which is incomplete, gives observed frequency, probability, expected frequency and chi-squared contribution. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 3.2}

   No. of defective ornaments in a week, \(r\)	Observed frequency	Probability	Expected frequency	Chi-squared contribution
   0	2	0.13534	5.4134	2.15232
   1	14
   2	13	0.27067		0.43620
   3	5		7.2179
   \(\geqslant 4\)	6	0.14288		0.01421

   \end{table}
   1. Complete the copy of the table in the Printed Answer Booklet.
   2. Carry out the test at the \(10 \%\) significance level.
3. On one occasion a fork-lift truck in the factory drops a crate containing eight ornaments and all of them are subsequently found to be defective. Explain why the Poisson model cannot model defects occurring in this manner.

7 A biologist is investigating migrating butterflies. Fig. 7.1 shows the numbers of migrating butterflies passing her location in 100 randomly chosen one-minute periods. \begin{table}[h] \end{table}

1. 1. Use the data to show that a suitable estimate for the mean number of butterflies passing her location per minute is 3.3.
   2. Explain how the value of the variance estimate calculated from the sample supports the suggestion that a Poisson distribution may be a suitable model for these data. The biologist decides to carry out a test to investigate whether a Poisson distribution may be a suitable model for these data.
2. In this question you must show detailed reasoning. Complete the copy of Fig. 7.2 of expected frequencies and contributions for a chi-squared test in the Printed Answer Booklet. \begin{table}[h]

   Number of butterflies	Frequency	Probability	Expected frequency	Chi-squared contribution
   0	6	0.0369	3.6883	1.4489
   1	9	0.1217	12.1714	0.8264
   2	18			0.2160
   3	26			0.6916
   4	13	0.1823	18.2252	1.4981
   5	16	0.1203	12.0286
   6	9	0.0662	6.6158	0.8593
   \(\geqslant 7\)	3	0.0510	5.0966	0.8625

   \captionsetup{labelformat=empty} \caption{Fig. 7.2}
   \end{table}
3. Complete the chi-squared test at the \(5 \%\) significance level.

5 A technician is investigating whether a batch of nylon 66 (a particular type of nylon) is contaminated by another type of nylon.
The average melting point of nylon 66 is \(264 ^ { \circ } \mathrm { C }\). However, if the batch is contaminated by the other type of nylon the melting point will be lower. The melting points, in \({ } ^ { \circ } \mathrm { C }\), of a random sample of 8 pieces of nylon from the batch are as follows. \(\begin{array} { l l l l l l l l } 262.7 & 265.0 & 264.1 & 261.7 & 262.9 & 263.5 & 261.3 & 262.6 \end{array}\)

1. Find
   The technician produces a Normal probability plot and carries out a Kolmogorov-Smirnov test for these data as shown in Fig. 5. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1db026a2-dffc-4877-b927-247fbf0e7a78-5_560_1358_982_246} \captionsetup{labelformat=empty} \caption{Fig. 5}
   \end{figure}
2. Comment on what the Normal probability plot and the \(p\)-value of the test suggest about the data.
3. In this question you must show detailed reasoning. Carry out a suitable test at the \(5 \%\) significance level to investigate whether the batch appears to be contaminated with another type of nylon.
4. Name an alternative test that could have been carried out if the population standard deviation had been known.

5 A particular alloy of bronze is specified as containing \(11.5 \%\) copper on average. A researcher takes a random sample of 14 specimens of this bronze and undertakes an analysis of each of them. The percentages of copper are found to be as follows. The researcher uses software to draw a Normal probability plot for these data and to conduct a Kolmogorov-Smirnov test for Normality. The output is shown in Fig 5.1. \begin{figure}[h] \end{figure}

1. Comment on what the Normal probability plot and the \(p\)-value of the test suggest about the data. The researcher uses software to produce a \(99 \%\) confidence interval for the mean percentage of copper in the alloy, based on the \(t\) distribution. The output from the software is shown in Fig 5.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0de8222f-7df5-4e17-ab68-0f9d84fc615d-5_1058_615_434_726} \captionsetup{labelformat=empty} \caption{Fig 5.2}
   \end{figure}
2. State the confidence interval which the software gives, in the form \(a < \mu < b\).
3. (A) State an assumption necessary for the use of the \(t\) distribution in the construction of this confidence interval.
   (B) State whether the assumption in part (iii) (A) seems reasonable.
4. Does the confidence interval suggest that the copper content is different from \(11.5 \%\), on average? Explain your answer.
5. In the output from the software shown in Fig 5.2, SE stands for 'standard error'.
   (A) Explain what a standard error is.
   (B) Show how the standard error was calculated in this case.
6. Suggest a way in which the researcher could produce a narrower confidence interval.

4 Zara uses a metal detector to search for coins on a beach.
She wonders if the numbers of coins that she finds in an area of \(10 \mathrm {~m} ^ { 2 }\) can be modelled by a Poisson distribution. The table below shows the numbers of coins that she finds in randomly chosen areas of \(10 \mathrm {~m} ^ { 2 }\) over a period of months.

1. Software gives the sample mean as 1.98 and the sample standard deviation as 1.4212. Explain how these values suggest that a Poisson distribution may be an appropriate model for the numbers of coins found. Zara decides to carry out a chi-squared test to investigate whether a Poisson distribution is an appropriate model.
   Fig. 4 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
   1	Number of coins found	Observed frequency	Expected frequency	Chi-squared contribution
   2	0	13	13.8069	0.0472
   3	1	28
   4	2	30	27.0643	0.3184
   5	3	14	17.8625	0.8352
   6	4	10	8.8419	0.1517
   7	\(\geqslant 5\)	5		0.0015

   \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{table}
2. Showing your calculations, find the missing values in each of the following cells.
   For the rest of this question, you should assume that the number of coins that Zara finds in an area of \(10 \mathrm {~m} ^ { 2 }\) can be modelled by a Poisson distribution with mean 1.98.
   Zara also finds pieces of jewellery independently of the coins she finds. The number of pieces of jewellery that she finds per \(10 \mathrm {~m} ^ { 2 }\) area is modelled by a Poisson distribution with mean 0.42 .
3. Find the probability that Zara finds a total of exactly 3 items (coins and/or jewellery) in an area of \(10 \mathrm {~m} ^ { 2 }\).
4. Find the probability that Zara finds a total of at least 30 items (coins and/or jewellery) in an area of \(100 \mathrm {~m} ^ { 2 }\).

3 Jane wonders whether the number of wasps entering a wasp's nest per 5 second interval can be modelled by a Poisson distribution with mean \(\mu\). She counts the number of wasps entering the nest over 60 randomly selected 5 -second intervals. The results are shown in Fig. 3.1. \begin{table}[h] \end{table}

1. Show that a suitable estimate for the value of \(\mu\) is 5.1. Fig. 3.2 shows part of a screenshot for a \(\chi ^ { 2 }\) test to assess the goodness of fit of a Poisson model. The sample mean has been used as an estimate for the population mean. Some of the values in the spreadsheet have been deliberately omitted. \begin{table}[h]

   	A	B	C	D	E
   \includegraphics[max width=\textwidth, alt={}]{e8624e9b-5143-49d2-9683-cc3a1082694e-4_132_40_1069_273}	Number of wasps	Observed frequency	Poisson probability	Expected frequency	Chi-squared contribution
   2	\(\leqslant 2\)	7	0.1165	6.9887	0.0000
   3	3	5		8.0874	1.1786
   4	4	12			0.2765
   5	5	10			0.0255
   6	6	10	0.1490	8.9400	0.1257
   7	7	11	0.1086	6.5134	3.0904
   8	\(\geqslant 8\)	5	0.1440	8.6414
   9

   \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{table}
2. Determine the missing values in each of the following cells, giving your answers correct to 4 decimal places.
   Carry out the hypothesis test at the 5\% significance level.
3. Jane also carries out a \(\chi ^ { 2 }\) test for the number of wasps leaving another nest. As part of her calculations, she finds that the probability of no wasps leaving the nest in a 5 -second period is 0.0053 . She finds that a Poisson distribution is also an appropriate model in this case. Find a suitable estimate for the value of the mean number of wasps leaving the nest per 5-second period.

4 Eve lives in a narrow lane in the country. She wonders whether the number of vehicles passing her house per minute can be modelled by a Poisson distribution with mean \(\mu\). She counts the number of vehicles passing her house over 100 randomly selected one-minute intervals. The results are shown in Table 4.1. \begin{table}[h] \end{table}

1. Use the results to find an estimate for \(\mu\). The spreadsheet in Fig. 4.2 shows data for a \(\chi ^ { 2 }\) test to assess the goodness of fit of a Poisson model. The sample mean from part (a) has been used as an estimate for the population mean. Some of the values in the spreadsheet have been deliberately omitted. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 4.2}

   \multirow[b]{2}{*}{1}	A	B	C	D	E
   	Number of vehicles	Observed frequency	Poisson probability	Expected frequency	Chi-squared contribution
   2	0	36	0.2725	27.2532	2.8073
   3	1	33	0.3543	35.4291
   4	2	14			3.5400
   5	\(\geqslant 3\)	17			0.5145
   6

   \end{table}
2. Calculate the missing values in each of the following cells, giving your answers correct to 4 decimal places.
   Carry out the \(\chi ^ { 2 }\) test at the 5\% significance level.
3. Eve checks her data and notices that the two largest numbers of vehicles per minute (8 and 10) occurred when some horses were being ridden along the lane, causing delays to the vehicles. She therefore repeats the analysis, missing out these two items of data. She finds that the value of the \(\chi ^ { 2 }\) test statistic is now 4.748. The number of degrees of freedom of the test is unchanged. Make two comments about this revised test.

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] \end{table}

1. Showing your calculations, find the missing values in each of the following cells.

9 A supermarket sells trays of peaches. Each tray contains 10 peaches. Often some of the peaches in a tray are rotten. The numbers of rotten peaches in a random sample of 150 trays are shown in Table 9.1. \begin{table}[h] \end{table} A manager at the supermarket thinks that the number of rotten peaches in a tray may be modelled by a binomial distribution.

1. Use these data to estimate the value of the parameter \(p\) for the binomial model \(\mathrm { B } ( 10 , p )\). The manager decides to carry out a goodness of fit test to investigate further. The screenshot in Fig. 9.2 shows part of a spreadsheet to assess the goodness of fit of the distribution \(\mathrm { B } ( 10 , p )\), using the value of \(p\) estimated from the data. \begin{table}[h]

   -	A	B	C	D	E
   1	Number of rotten peaches	Observed frequency	Binomial probability	Expected frequency	Chi-squared contribution
   2	0	39
   3	1	39			1.4229
   4	2	33	0.2941	44.1167	2.8012
   5	3	19	0.1629	24.4383	1.2102
   6	\(\geqslant 4\)	20	0.0769	11.5311	6.2199
   7

   \captionsetup{labelformat=empty} \caption{Fig. 9.2}
   \end{table}
2. Calculate the missing values in each of the following cells.

6 Cosmic rays passing through the upper atmosphere cause muons, and other types of particle, to be formed. Muons can be detected when they reach the surface of the earth. It is known that the mean number of muons reaching a particular detector is 1.7 per second. The numbers of muons reaching this detector in 200 randomly selected periods of 1 second are shown in Fig. 6.1. \begin{table}[h] \end{table}

1. Use the values of the sample mean and sample variance to discuss the suitability of a Poisson distribution as a model. The screenshot in Fig. 6.2 shows part of a spreadsheet to assess the goodness of fit of the distribution Po(1.7). \begin{table}[h]

   	A	B	C	D	E
   1	Number of muons	Observed frequency	Poisson probability	Expected frequency	Chi-squared contribution
   2	0	34	0.1827	36.5367	0.1761
   3	1	65
   4	2	55	0.2640	52.7955	0.0920
   5	3	24	0.1496	29.9175	1.1704
   6	4	14			0.1299
   7	\(\geqslant 5\)	8	0.0296	5.9230	0.7284

   \captionsetup{labelformat=empty} \caption{Fig. 6.2}
   \end{table}
2. Calculate the missing values in each of the following cells.
   Carry out the test at the 5\% significance level.

5. Chris is investigating the distribution of birth months for ice hockey players. He collects data for 869 randomly chosen National Hockey League (NHL) players. He decides to carry out a chi-squared test. Using a spreadsheet, he produces the following output.

1. By considering the output, state the null hypothesis that Chris is testing. State what conclusion Chris should reach and explain why. Chris now wonders if Premier League football players' birth months are distributed uniformly throughout the year. He collects the birth months of 75 randomly selected Premier League footballers. This information is shown in the table below.

   Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
   3	7	11	4	12	2	6	6	5	8	5	6

2. Carry out the chi-squared goodness of fit test at the 10\% significance level that Chris should use to conduct his investigation.

5. John has a game that involves throwing a set of three identical, cubical dice with faces numbered 1 to 6 . He wishes to investigate whether these dice are fair in terms of the number of sixes obtained when they are thrown. John throws the set of three dice 1100 times and records the number of sixes obtained for each throw. The results are shown in the table below. Using these results, conduct a goodness of fit test and draw an appropriate conclusion.

1. A company makes bags. The table below shows the number of bags sold on a random sample of 50 days. A manager believes that the number of bags sold per day can be modelled by the Poisson distribution with mean \(2 \cdot 2\).

1. Carry out a chi-squared goodness of fit test, using a \(10 \%\) significance level.
2. A chi-squared goodness of fit test for the Poisson distribution with mean \(2 \cdot 5\) is conducted. This uses the same number of degrees of freedom as part (a) and gives a test statistic of 1.53 . State, with a reason, which of these two Poisson models is a better fit for the data.

1. A researcher is investigating the distribution of orchids in a field. He believes that the Poisson distribution with a mean of 1.75 may be a good model for the number of orchids in each square metre. He randomly selects 150 non-overlapping areas, each of one square metre, and counts the number of orchids present in each square.

The results are recorded in the table below. He calculates the expected frequencies as follows

1. Find the value of \(r\) giving your answer to 2 decimal places. The researcher will test, at the \(5 \%\) level of significance, whether or not the data can be modelled by a Poisson distribution with mean 1.75
2. State clearly the hypotheses required to test whether or not this Poisson distribution is a suitable model for these data. The test statistic for this test is 2.0 and the number of degrees of freedom to be used is 4
3. Explain fully why there are 4 degrees of freedom.
4. Stating your critical value clearly, determine whether or not these data support the researcher's belief. The researcher works in another field where the number of orchids in each square metre is known to have a Poisson distribution with mean 1.5 He randomly selects 200 non-overlapping areas, each of one square metre, in this second field, and counts the number of orchids present in each square.
5. Using a Poisson approximation, show that the probability that he finds at least one square with exactly 6 orchids in it is 0.506 to 3 decimal places.

1. A spinner used for a game is designed to give scores with the following probabilities

The spinner is spun 80 times and the results are as follows Test, at the \(10 \%\) level of significance, whether or not the spinner is giving scores as it is designed to do. Show your working and state your hypotheses clearly.

1. In an experiment, James flips a coin 3 times and records the number of heads. He carries out the experiment 100 times with his left hand and 100 times with his right hand.

1. Test, at the \(5 \%\) level of significance, whether or not there is an association between the hand he flips the coin with and the number of heads. You should state your hypotheses, the degrees of freedom and the critical value used for this test.
2. Assuming the coin is unbiased, write down the distribution of the number of heads in 3 flips.
3. Carry out a \(\chi ^ { 2 }\) test, at the \(10 \%\) level of significance, to test whether or not the distribution you wrote down in part (b) is a suitable model for the number of heads obtained in the 200 trials of James' experiment. You should state your hypotheses, the degrees of freedom and the critical value used for this test.

1. Flobee sells tomato seeds in packets, each containing 40 seeds. Flobee advertises that only 4\% of its tomato seeds do not germinate.

Amodita is investigating the germination of Flobee's tomato seeds. She plants 125 packets of Flobee's tomato seeds and records the number of seeds that do not germinate in each packet. Amodita wants to test whether the binomial distribution \(\mathrm { B } ( 40,0.04 )\) is a suitable model for these data. The table below shows the expected frequencies, to 2 decimal places, using this model.

1. Calculate the value of \(r\) and the value of \(s\)
2. Stating your hypotheses clearly, carry out the test at the \(5 \%\) level of significance. You should state the number of degrees of freedom, critical value and conclusion clearly. Amodita believes that Flobee should use a more realistic value for the percentage of their tomato seeds that do not germinate.
   She decides to test the data using a new model \(\mathrm { B } ( 40 , p )\)
3. Showing your working, suggest a more realistic value for \(p\)