Chi-squared distribution

7 Mohammed is conducting a medical trial to study the effect of two drugs, \(A\) and \(B\), on the amount of time it takes to recover from a particular illness. Drug \(A\) is used by one group of 60 patients and drug \(B\) is used by a second group of 60 patients. The results are summarised in the table:

2 A test for association is to be carried out. The tables below show the observed frequencies and the expected frequencies that are to be used for the test. It is necessary to merge some rows or columns before the test can be carried out.
Find the entry in the tables that provides evidence for this.
Circle your answer.
[0pt] [1 mark]
Observed A-Z
Observed B-Z
Expected A-X
Expected B-Y

A researcher is investigating the relationship between the political allegiance of university students and their childhood environment. He chooses a random sample of 100 students and finds that 60 have political allegiance to the Alliance party. He also classifies their childhood environment as rural or urban, and finds that 45 had a rural childhood. The researcher carries out a test, at the \(10 \%\) significance level, on this data and finds that political allegiance is independent of childhood environment. Given that \(A\) is the number of students in the sample who both support the Alliance party and have a rural childhood, find the greatest and least possible values of \(A\). A second random sample of size \(100 N\), where \(N\) is an integer, is taken from the university student population. It is found that the proportions supporting the Alliance party from urban and rural childhoods are the same as in the first sample. Given that the value of \(A\) in the first sample was 29, find the greatest possible value of \(N\) that would lead to the same conclusion (that political allegiance is independent of childhood environment) from a test, at the \(10 \%\) significance level, on this second set of data.

8 Drinking glasses are sold in packs of 4. The manufacturer conducts a survey to assess the quality of the glasses. The results from a sample of 50 randomly chosen packs are summarised in the following table. Fit a binomial distribution to the data and carry out a goodness of fit test at the \(10 \%\) significance level.

3 Apples are sold in bags of 5. Based on her previous experience, Freya claims that the probability of any apple weighing more than 100 grams is 0.35 , independently of other apples in the bag. The apples in a random sample of 150 bags are checked and the number, \(x\), in each bag weighing more than 100 grams is recorded. The results are shown in the following table. Carry out a goodness of fit test at the \(5 \%\) significance level and hence comment on Freya's claim.

Each of 200 identically biased dice is thrown repeatedly until an even number is obtained. The number of throws, \(x\), needed is recorded and the results are summarised in the following table. State a type of distribution that could be used to fit the data given in the table above. Fit a distribution of this type in which the probability of throwing an even number for each die is 0.6 and carry out a goodness of fit test at the 5\% significance level. For each of these dice, it is known that the probability of obtaining a 6 when it is thrown is 0.25 . Ten of these dice are each thrown 5 times. Find the probability that at least one 6 is obtained on exactly 4 of the 10 dice.

8 The number of goals scored by a certain football team was recorded for each of 100 matches, and the results are summarised in the following table. Fit a Poisson distribution to the data, and test its goodness of fit at the 5\% significance level.

16
256 2 The random variable \(T\) has an exponential distribution with mean 2 Find \(\mathrm { P } ( T \leq 1.4 )\) Circle your answer. \(\mathrm { e } ^ { - 2.8 }\) \(\mathrm { e } ^ { - 0.7 }\) \(1 - e ^ { - 0.7 }\) \(1 - \mathrm { e } ^ { - 2.8 }\) The continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 2 \\ - \frac { 1 } { 9 } y ^ { 2 } + \frac { 10 } { 9 } y - \frac { 16 } { 9 } & 2 \leq y < 5 \\ 1 & y \geq 5 \end{array} \right.$$ Find the median of \(Y\) Circle your answer. 2 \(\frac { 10 - 3 \sqrt { 2 } } { 2 }\) \(\frac { 7 } { 2 }\) \(\frac { 10 + 3 \sqrt { 2 } } { 2 }\) Turn over for the next question 4 Research has shown that the mean number of volcanic eruptions on Earth each day is 20 Sandra records 162 volcanic eruptions during a period of one week. Sandra claims that there has been an increase in the mean number of volcanic eruptions per week. Test Sandra's claim at the \(5 \%\) level of significance.
5 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 6 } e ^ { \frac { x } { 3 } } & 0 \leq x \leq \ln 27 \\ 0 & \text { otherwise } \end{cases}$$ Show that the mean of \(X\) is \(\frac { 3 } { 2 } ( \ln 27 - 2 )\) 6 Over time it has been accepted that the mean retirement age for professional baseball players is 29.5 years old. Imran claims that the mean retirement age is no longer 29.5 years old.
He takes a random sample of 5 recently retired professional baseball players and records their retirement ages, \(x\). The results are $$\sum x = 152.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 7.81$$ 6

1. State an assumption that you should make about the distribution of the retirement ages to investigate Imran's claim. 6
2. Investigate Imran's claim, using the 10\% level of significance.

8 The number of goals scored by a certain football team was recorded for each of 100 matches, and the results are summarised in the following table. Fit a Poisson distribution to the data, and test its goodness of fit at the 5\% significance level.

7 A random sample of 80 observations of the continuous random variable \(X\) was taken and the values are summarised in the following table. It is required to test the goodness of fit of the distribution having probability density function f given by $$f ( x ) = \begin{cases} \frac { 3 } { x ^ { 2 } } & 2 \leqslant x < 6 \\ 0 & \text { otherwise. } \end{cases}$$ Show that the expected frequency for the interval \(2 \leqslant x < 3\) is 40 and calculate the remaining expected frequencies. Carry out a goodness of fit test, at the \(10 \%\) significance level.

5 A statistician suggested that the weekly sales \(X\) thousand litres at a petrol station could be modelled by the following probability density function. $$f ( x ) = \begin{cases} \frac { 1 } { 40 } ( 2 x + 3 ) & 0 \leqslant x < 5 \\ 0 & \text { otherwise } \end{cases}$$

1. Show that, using this model, \(\mathrm { P } ( a \leqslant X < a + 1 ) = \frac { a + 2 } { 20 }\) for \(0 \leqslant a \leqslant 4\). Sales in 100 randomly chosen weeks gave the following grouped frequency table.

   \(x\)	\(0 \leqslant x < 1\)	\(1 \leqslant x < 2\)	\(2 \leqslant x < 3\)	\(3 \leqslant x < 4\)	\(4 \leqslant x < 5\)
   Frequency	16	12	18	30	24

2. Carry out a goodness of fit test at the \(10 \%\) significance level of whether \(\mathrm { f } ( x )\) fits the data.

3 The random variable \(X\) has the following probability density function, \(\mathrm { f } ( x )\). $$f ( x ) = \begin{cases} k x ( x - 5 ) ^ { 2 } & 0 \leqslant x < 5 \\ 0 & \text { elsewhere } \end{cases}$$

1. Sketch \(\mathrm { f } ( x )\).
2. Find, in terms of \(k\), the cumulative distribution function, \(\mathrm { F } ( x )\).
3. Hence show that \(k = \frac { 12 } { 625 }\). The random variable \(X\) is proposed as a model for the amount of time, in minutes, lost due to stoppages during a football match. The times lost in a random sample of 60 matches are summarised in the table. The table also shows some of the corresponding expected frequencies given by the model.

   Time (minutes)	\(0 \leqslant x < 1\)	\(1 \leqslant x < 2\)	\(2 \leqslant x < 3\)	\(3 \leqslant x < 4\)	\(4 \leqslant x < 5\)
   Observed frequency	5	15	23	11	6
   Expected frequency			17.76	9.12	1.632

4. Find the remaining expected frequencies.
5. Carry out a goodness of fit test, using a significance level of \(2.5 \%\), to see if the model might be suitable in this context.

2 The discrete random variable \(Y\) is uniformly distributed over the values \(\{ 12,13 , \ldots , 20 \}\).

1. Write down \(\mathrm { P } ( Y < 15 )\).
2. Two independent observations of \(Y\) are taken. Find the probability that one of these values is less than 15 and the other is greater than 15 .
3. Find \(\mathrm { P } ( Y > \mathrm { E } ( Y ) )\).

2 The discrete random variable \(Y\) is uniformly distributed over the values \(\{ 12,13 , \ldots , 20 \}\).

1. Write down \(\mathrm { P } ( Y < 15 )\).
2. Two independent observations of \(Y\) are taken. Find the probability that one of these values is less than 15 and the other is greater than 15 .
3. Find \(\mathrm { P } ( Y > \mathrm { E } ( Y ) )\).

8 The table shows the results of a random sample drawn from a population which is thought to have the distribution \(\mathrm { U } ( 20 )\). Find the range of values of \(y\) for which the data are not consistent with the distribution at the \(5 \%\) significance level. \section*{END OF QUESTION PAPER}

2 In a study of the inheritance of skin colouration in corn snakes, a researcher found 865 snakes with black and orange bodies, 320 snakes with black bodies, 335 snakes with orange bodies and 112 snakes with bodies of other colours. Theory predicts that snakes of these colours should occur in the ratios \(9 : 3 : 3 : 1\). Test, at the \(5 \%\) significance level, whether these experimental results are compatible with theory.

2 In a study of the inheritance of skin colouration in corn snakes, a researcher found 865 snakes with black and orange bodies, 320 snakes with black bodies, 335 snakes with orange bodies and 112 snakes with bodies of other colours. Theory predicts that snakes of these colours should occur in the ratios \(9 : 3 : 3 : 1\). Test, at the \(5 \%\) significance level, whether these experimental results are compatible with theory.

7 Benford's Law states that, in many tables containing large numbers of numerical values, the probability distribution of the leading non-zero digit \(D\) is given by $$\mathrm { P } ( D = d ) = \log _ { 10 } \left( \frac { d + 1 } { d } \right) , \quad d = 1,2 , \ldots , 9 .$$ The following table shows a summary of a random sample of 100 non-zero leading digits taken from a table of cumulative probabilities for the Poisson distribution. Carry out a suitable goodness of fit test at the 10\% significance level.

2. The standard deviation of the length of a random sample of 8 fence posts produced by a timber yard was 8 mm . A second timber yard produced a random sample of 13 fence posts with a standard deviation of 14 mm .

1. Test, at the \(10 \%\) significance level, whether or not there is evidence that the lengths of fence posts produced by these timber yards differ in variability. State your hypotheses clearly.
2. State an assumption you have made in order to carry out the test in part (a).

7. A psychologist gives a test to students from two different schools, \(A\) and \(B\). A group of 9 students is randomly selected from school \(A\) and given instructions on how to do the test.
A group of 7 students is randomly selected from school \(B\) and given the test without the instructions. The table shows the time taken, to the nearest second, to complete the test by the two groups. Stating your hypotheses clearly,

1. test at the \(10 \%\) significance level, whether or not the variance of the times taken to complete the test by students from school \(A\) is the same as the variance of the times taken to complete the test by students from school \(B\). (You may assume that times taken for each school are normally distributed.)
2. test at the \(5 \%\) significance level, whether or not the mean time taken to complete the test by students from school \(A\) is greater than the mean time taken to complete the test by students from school \(B\).
3. Why does the result to part (a) enable you to carry out the test in part (b)?
4. Give one factor that has not been taken into account in your analysis.

2. The weights, in grams, of apples are assumed to follow a normal distribution. The weights of apples sold by a supermarket have variance \(\sigma _ { s } { } ^ { 2 }\). A random sample of 4 apples from the supermarket had weights $$\text { 114, 100, 119, } 123 .$$

1. Find a 95\% confidence interval for \(\sigma _ { s } ^ { 2 }\). The weights of apples sold on a market stall have variance \(\sigma _ { M } ^ { 2 }\). A second random sample of 7 apples was taken from the market stall. The sample variance \(s _ { M } ^ { 2 }\) of the apples was 318.8.
2. Stating your hypotheses clearly test, at the \(1 \%\) levcel of significnace, whether or not there is evidence that \(\sigma _ { M } ^ { 2 } > \sigma _ { s } ^ { 2 }\).

8 The continuous random variable \(Y\) has a distribution with mean \(\mu\) and variance 20. A random sample of 50 observations of \(Y\) is selected and these observations are summarised in the following grouped frequency table.

1. Assuming that \(Y \sim \mathrm {~N} ( 25,20 )\), show that the expected frequency for the interval \(20 \leqslant y < 25\) is 18.41, correct to 2 decimal places, and obtain the remaining expected frequencies.
2. Test, at the \(5 \%\) significance level, whether the distribution \(\mathrm { N } ( 25,20 )\) fits the data.
3. Given that the sample mean is 24.91 , find a \(98 \%\) confidence interval for \(\mu\).
4. Does the outcome of the test in part (ii) affect the validity of the confidence interval found in part (iii)? Justify your answer.

2 It is claimed that the heights of a particular age group of boys follow a normal distribution with mean 125 cm and standard deviation 12 cm . Observations for a randomly chosen group of 60 boys in this age group are summarised in the following table. The table also gives the expected frequencies, correct to 2 decimal places, based on the normal distribution with mean 125 cm and standard deviation 12 cm .

1. Show how the expected frequency for \(130 \leqslant x < 140\) is obtained.
2. Carry out a goodness of fit test, at the \(5 \%\) significance level, to determine whether the claim is supported by the data.

2 It is claimed that the heights of a particular age group of boys follow a normal distribution with mean 125 cm and standard deviation 12 cm . Observations for a randomly chosen group of 60 boys in this age group are summarised in the following table. The table also gives the expected frequencies, correct to 2 decimal places, based on the normal distribution with mean 125 cm and standard deviation 12 cm .

1. Show how the expected frequency for \(130 \leqslant x < 140\) is obtained.
2. Carry out a goodness of fit test, at the \(5 \%\) significance level, to determine whether the claim is supported by the data.

4. A random sample of 15 tomatoes is taken and the weight \(x\) grams of each tomato is found. The results are summarised by \(\sum x = 208\) and \(\sum x ^ { 2 } = 2962\).

1. Assuming that the weights of the tomatoes are normally distributed, calculate the \(90 \%\) confidence interval for the variance \(\sigma ^ { 2 }\) of the weights of the tomatoes.
2. State with a reason whether or not the confidence interval supports the assertion \(\sigma ^ { 2 } = 3\).

1. A diabetic patient records her blood glucose readings in \(\mathrm { mmol } / \mathrm { l }\) at random times of day over several days. Her readings are given below.

$$\begin{array} { l l l l l l l } 5.3 & 5.7 & 8.4 & 8.7 & 6.3 & 8.0 & 7.2 \end{array}$$ Assuming that the blood glucose readings are normally distributed calculate

1. an unbiased estimate for the variance \(\sigma ^ { 2 }\) of the blood glucose readings,
2. a \(90 \%\) confidence interval for the variance \(\sigma ^ { 2 }\) of blood glucose readings.
3. State whether or not the confidence interval supports the assertion that \(\sigma = 0.9\). Give a reason for your answer.

7. The times taken to travel to school by sixth form students are normally distributed. A head teacher records the times taken to travel to school, in minutes, of a random sample of 10 sixth form students from her school. Based on this sample, the \(95 \%\) confidence interval for the mean time taken to travel to school for sixth form students from her school is
[0pt] [28.5, 48.7] Calculate a 99\% confidence interval for the variance of the time taken to travel to school for sixth form students from her school.
(9)

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] \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] \end{table}

1. Complete Table 2
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.

1. The random variable \(X\) has an \(F\)-distribution with 8 and 12 degrees of freedom.

Find \(\mathrm { P } \left( \frac { 1 } { 5.67 } < X < 2.85 \right)\).
(4)

7 The discrete random variable \(X\) is equally likely to take values 0,1 and 2 . \(3 N\) observations of \(X\) are obtained, and the observed frequencies corresponding to \(X = 0 , X = 1\) and \(X = 2\) are given in the following table. The test statistic for a chi-squared goodness of fit test for the data is 0.3 . Find the value of \(N\).

7 A restaurant has asked Sylvia to conduct a \(\chi ^ { 2 }\) test for association between meal ordered and age of customer. 7

1. State the hypotheses that Sylvia should use for her test. 7
2. Sylvia correctly calculates her value of the test statistic to be 44.1
   She uses a \(5 \%\) level of significance and the degrees of freedom for the test is 30
   Sylvia accepts the null hypothesis.
   Explain whether or not Sylvia was correct to accept the null hypothesis.
   7
3. State in context the correct conclusion to Sylvia's test.

3 A mobile phone company offers an insurance policy to its customers when they purchase a mobile phone. The company conducted a survey on the age of the customers and whether or not claims were made. A random sample of 1200 customers from this company was investigated for 2020 and the results are shown in the table below. The data are to be used to determine whether or not making a claim is independent of age.

1. Calculate the expected frequencies for the age group 51 years and over that
   1. made a claim in 2020
   2. did not make a claim in 2020 The 4 classes of customers aged between 17 and 50 give a value of \(\sum \frac { ( O - E ) ^ { 2 } } { E } = 7.123\) correct to 3 decimal places.
2. Test, at the \(1 \%\) level of significance, whether or not making a claim is independent of age. Show your working clearly, stating your hypotheses, the degrees of freedom, the test statistic and the critical value used.