5.06b Fit prescribed distribution: chi-squared test

4 A random variable \(X\) has probability density function \(\mathrm { f } ( x ) = \frac { 2 x } { \lambda ^ { 2 } }\) for \(0 < x < \lambda\), where \(\lambda\) is a positive constant.

1. Show that, for any value of \(\lambda , \mathrm { f } ( x )\) is a valid probability density function.
2. Find \(\mu\), the mean value of \(X\), in terms of \(\lambda\) and show that \(\mathrm { P } ( X < \mu )\) does not depend on \(\lambda\).
3. Given that \(\mathrm { E } \left( X ^ { 2 } \right) = \frac { \lambda ^ { 2 } } { 2 }\), find \(\sigma ^ { 2 }\), the variance of \(X\), in terms of \(\lambda\). The random variable \(X\) is used to model the depth of the space left by the filling machine at the top of a jar of jam. The model gives the following probabilities for \(X\) (whatever the value of \(\lambda\) ).

   \(0 < X \leqslant \mu - \sigma\)	\(\mu - \sigma < X \leqslant \mu\)	\(\mu < X \leqslant \mu + \sigma\)	\(\mu + \sigma < X < \lambda\)
   0.18573	0.25871	0.36983	0.18573

   A sample of 50 random observations of \(X\), classified in the same way, is summarised by the following frequencies.

   4

   11

   20

   15

4. Carry out a suitable test at the \(5 \%\) level of significance to assess the goodness of fit of \(X\) to these data. Explain briefly how your conclusion may be affected by the choice of significance level.

4 The numbers of call-outs per day received by a fire station for a random sample of 255 weekdays were recorded as follows. The mean number of call-outs per day for these data is 0.6 . A Poisson model, using this sample mean of 0.6 , is fitted to the data, and gives the following expected frequencies (correct to 3 decimal places).

1. Using a \(5 \%\) significance level, carry out a test to examine the goodness of fit of the model to the data. The time \(T\), measured in days, that elapses between successive call-outs can be modelled using the exponential distribution for which \(\mathrm { f } ( t )\), the probability density function, is $$\mathrm { f } ( t ) = \begin{cases} 0 & t < 0 , \\ \lambda \mathrm { e } ^ { - \lambda t } & t \geqslant 0 , \end{cases}$$ where \(\lambda\) is a positive constant.
2. For the distribution above, it can be shown that \(\mathrm { E } ( T ) = \frac { 1 } { \lambda }\). Given that the mean time between successive call-outs is \(\frac { 5 } { 3 }\) days, write down the value of \(\lambda\).
3. Find \(\mathrm { F } ( t )\), the cumulative distribution function.
4. Find the probability that the time between successive call-outs is more than 1 day.
5. Find the median time that elapses between successive call-outs.

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.

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.

A family was asked to record the number of letters delivered to their house on each of 200 randomly chosen weekdays. The results are summarised in the following table. It is suggested that the number of letters delivered each weekday has a Poisson distribution. By finding the mean and variance for this sample, comment on the appropriateness of this suggestion. The following table includes some of the expected values, correct to 3 decimal places, using a Poisson distribution with mean equal to the sample mean for the above data.

1. Show that \(p = 46.304\), correct to 3 decimal places, and find \(q\).
2. Carry out a goodness of fit test at the \(10 \%\) 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.

9 A random sample of 200 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 with probability density function \(f\) given by $$f ( x ) = \begin{cases} \frac { 1 } { x \ln 8 } & 1 \leqslant x < 8 \\ 0 & \text { otherwise } \end{cases}$$ The relevant expected frequencies, correct to 2 decimal places, are given in the following table. Show that \(p = 39.00\), correct to 2 decimal places, and find the value of \(q\). Carry out a goodness of fit test at the 5\% significance level.

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.

9 Applicants for a national teacher training course are required to pass a mathematics test. Each year, the applicants are tested in groups of 6 and the number of successful applicants in each group is recorded. The overall proportion of successful applicants has remained constant over the years and is equal to \(60 \%\) of the applicants. The results from 150 randomly chosen groups are shown in the following table. Test, at the \(5 \%\) significance level, the goodness of fit of the distribution \(\mathbf { B } ( 6,0.6 )\) for the number of successful applicants in a group.

A scientist carries out an experiment to investigate the quantity \(X\), which takes the values \(0,1,2,3,4\), 5 or 6 . He believes that the values taken by \(X\) follow a binomial distribution. He conducts 250 trials. His results are summarised in the following table.

1. Show that unbiased estimates of the mean and variance for these results are 1.876 and 1.266 respectively, correct to 3 decimal places. By evaluating the mean and variance of the distribution B(6, 0.313), explain why \(X\) could have this distribution.
   The expected frequencies corresponding to the distribution \(\mathrm { B } ( 6,0.313 )\) are shown in the following table.

   \(x\)	0	1	2	3	4	5	6
   Observed frequency	22	83	72	53	17	3	0
   Expected frequency	26.3	71.9	81.8	49.7	17.0	3.1	0.2

2. Show how the expected frequency for \(x = 4\) is calculated.
3. Test at the \(5 \%\) significance level whether the scientist's belief is correct.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

9 A random sample of 50 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 with probability density function \(f\) given by $$f ( x ) = \begin{cases} \frac { 3 } { 16 } ( 4 - x ) ^ { \frac { 1 } { 2 } } & 0 \leqslant x < 4 \\ 0 & \text { otherwise. } \end{cases}$$ The relevant expected frequencies, correct to 2 decimal places, are given in the following table.

1. Show how the expected frequency for \(1.6 \leqslant x < 2.4\) is obtained.
2. Carry out a goodness of fit test at the \(5 \%\) significance level.

9 A sample of 100 observations of the continuous random variable \(T\) was obtained and the values are summarised in the following table. It is required to test the goodness of fit of the distribution with probability density function given by $$f ( t ) = \begin{cases} \frac { 9 } { 4 t ^ { 3 } } & 1 \leqslant t < 3 \\ 0 & \text { otherwise } \end{cases}$$ The relevant expected values are as follows. Show how the expected value 10.125 is obtained. Carry out the test, at the \(10 \%\) significance level.

8 A sample of 216 observations of the continuous random variable \(X\) was obtained and the results are summarised in the following table. It is suggested that these results are consistent with a distribution having probability density function f given by $$f ( x ) = \begin{cases} k x ^ { 2 } & 0 \leqslant x < 6 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant. The relevant expected frequencies are given in the following table.

1. Show that \(a = 19\) and find the values of \(b\) and \(c\).
2. Carry out a goodness of fit test at the \(10 \%\) significance level.

A continuous random variable \(X\) is believed to have the probability density function f given by $$f ( x ) = \begin{cases} \frac { 3 } { 10 } \left( 5 x - x ^ { 2 } - 4 \right) & 2 \leqslant x < 4 \\ 0 & \text { otherwise } \end{cases}$$ A random sample of 60 observations was taken and these values are summarised in the following grouped frequency table. The estimated mean, based on the grouped data in the table above, is 2.69 , correct to 2 decimal places. It is decided that a goodness of fit test will only be conducted if the mean predicted from the probability density function is within \(10 \%\) of the estimated mean. Show that this condition is satisfied. The relevant expected frequencies are as follows. Show how the expected frequency for the interval \(3.2 \leqslant x < 3.6\) is obtained. Carry out the goodness of fit test at the 10\% significance level.

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.

8 A factory produces china mugs. Random samples of size 6 are selected at regular intervals, and the mugs are inspected for defects. During one week, 100 samples are selected and the numbers of defective mugs found are summarised in the following table. Fit a binomial distribution to the data and carry out a goodness of fit test at the 5\% significance level.

8 The numbers of a particular type of laptop computer sold by a store on each of 100 consecutive Saturdays are summarised in the following table. Fit a Poisson distribution to the data and carry out a goodness of fit test at the \(2.5 \%\) significance level.

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.

9 The number of visitors arriving at an art exhibition is recorded for each 10 -minute period of time during the ten hours that it is open on a particular day. The results are as follows.

1. Calculate the mean and variance for this sample and explain whether your answers support a suggestion that a Poisson distribution might be a suitable model for the number of visitors in a 10-minute period.
2. Use an appropriate Poisson distribution to find the two expected frequencies missing from the following table.

   Number of visitors in a 10-minute period	0	1	2	3	4	5	6	7	8	\(\geqslant 9\)
   Expected number of 10 -minute periods	1.10		8.79		11.72	9.38	6.25	3.57	1.79	1.28

3. Test, at the \(10 \%\) significance level, the goodness of fit of this Poisson distribution to the data.

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.

4

1. In Germany, towards the end of the nineteenth century, a study was undertaken into the distribution of the sexes in families of various sizes. The table shows some data about the numbers of girls in 500 families, each with 5 children. It is thought that the binomial distribution \(\mathrm { B } ( 5 , p )\) should model these data.

   Number of girls	Number of families
   0	32
   1	110
   2	154
   3	125
   4	63
   5	16

   1. Use this information to calculate an estimate for the mean number of girls per family of 5 children. Hence show that 0.45 can be taken as an estimate of \(p\).
   2. Investigate at a \(5 \%\) significance level whether the binomial model with \(p\) estimated as 0.45 fits the data. Comment on your findings and also on the extent to which the conditions for a binomial model are likely to be met.
2. A researcher wishes to select 50 families from the 500 in part (a) for further study. Suggest what sort of sample she might choose and describe how she should go about choosing it.

13 At a certain factory Christmas tree decorations are packed in boxes of 10 . The quality control manager collects a random sample of 100 boxes of decorations and records the number of decorations in each box which are damaged. His results are displayed in Fig. 13.1. \begin{table}[h] \end{table}

1. Calculate
   It is believed that the number of damaged decorations in a box of 10, \(X\), may be modelled by a binomial distribution such that \(\mathrm { X } \sim \mathrm { B } ( \mathrm { n } , \mathrm { p } )\).
2. State suitable values for \(n\) and \(p\).
3. Use the binomial model to complete the copy of Fig. 13.2 in the Printed Answer Booklet, giving your answers correct to \(\mathbf { 1 }\) decimal place. \begin{table}[h]

   Number of damaged decorations	0	1	2	3	4	5 or more
   Observed number of boxes	19	35	28	13	5	0
   Expected number of boxes

   \captionsetup{labelformat=empty} \caption{Fig. 13.2}
   \end{table}
4. Explain whether the model is a good fit for these data.

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}

6 A student believes that if you ask people to choose an integer between 1 and 10, not all integers are equally likely to be chosen. The student asks a random sample of 100 people to choose an integer between 1 and 10 inclusive. The observed frequencies \(O\), together with the values of \(\frac { ( O - E ) ^ { 2 } } { E }\) where \(E\) is the corresponding expected frequency, are shown in the table.

1. Show how the value of 8.1 for integer 7 is obtained.
2. Show that there is evidence at the \(1 \%\) significance level that the student's belief is correct. The student wishes to suggest an alternative model for the probabilities associated with each integer. In this model, two of the integers have the same probability \(p _ { 1 }\) of being chosen and the other eight integers each have probability \(p _ { 2 }\) of being chosen.
3. Suggest which two integers should have probability \(p _ { 1 }\) and suggest a possible value of \(p _ { 1 }\).

9 The head teacher of a school believes that, on average, pupil absences on the days Monday, Tuesday, Wednesday, Thursday and Friday are in the ratio \(3 : 2 : 2 : 2 : 3\). The head teacher takes a random sample of 120 pupil absences. The results are as follows.

1. Test at the \(5 \%\) significance level whether these results are consistent with the head teacher's belief. A significance test at the \(5 \%\) level is also carried out on a second, independent, random sample of \(n\) pupil absences. All the numbers of absences are integers. The ratio of the numbers of absences for each day in this sample is identical to the ratio of the numbers of absences for each day in the original sample of size 120.
2. Determine the smallest value of \(n\) for which the conclusion of this significance test is that the data are not consistent with the head teacher's belief.