Chi-squared goodness of fit: Normal

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.

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.

4

1. An amateur weather forecaster has been keeping records of air pressure, measured in atmospheres. She takes the measurement at the same time every day using a barometer situated in her garden. A random sample of 100 of her observations is summarised in the table below. The corresponding expected frequencies for a Normal distribution, with its two parameters estimated by sample statistics, are also shown in the table.

   Pressure ( \(a\) atmospheres)	Observed frequency	Frequency as given by Normal model
   \(a \leqslant 0.98\)	4	1.45
   \(0.98 < a \leqslant 0.99\)	6	5.23
   \(0.99 < a \leqslant 1.00\)	9	13.98
   \(1.00 < a \leqslant 1.01\)	15	23.91
   \(1.01 < a \leqslant 1.02\)	37	26.15
   \(1.02 < a \leqslant 1.03\)	21	18.29
   \(1.03 < a\)	8	10.99

   Carry out a test at the \(5 \%\) level of significance of the goodness of fit of the Normal model. State your conclusion carefully and comment on your findings.
2. The forecaster buys a new digital barometer that can be linked to her computer for easier recording of observations. She decides that she wishes to compare the readings of the new barometer with those of the old one. For a random sample of 10 days, the readings (in atmospheres) of the two barometers are shown below.

   Day	A	B	C	D	E	F	G	H	I	J
   Old	0.992	1.005	1.001	1.011	1.026	0.980	1.020	1.025	1.042	1.009
   New	0.985	1.003	1.002	1.014	1.022	0.988	1.030	1.016	1.047	1.025

   Use an appropriate Wilcoxon test to examine at the \(10 \%\) level of significance whether there is any reason to suppose that, on the whole, readings on the old and new barometers do not agree.

7 In 1761, James Short took measurements of the parallax of the sun based on the transit of Venus. The mean and standard deviation of a random sample of 50 of these measurements are 8.592 and 0.7534 respectively, in suitable units.

1. Show that if \(X \sim \mathrm {~N} \left( 8.592,0.7534 ^ { 2 } \right)\), then $$\mathrm { P } ( X \leqslant 8.084 ) = \mathrm { P } ( 8.084 < X \leqslant 8.592 ) = \mathrm { P } ( 8.592 < X \leqslant 9.100 ) = \mathrm { P } ( X > 9.100 ) = 0.25 \text {. }$$ The following table summarises the 50 measurements using these intervals.

   Measurement \(( x )\)	\(x \leqslant 8.084\)	\(8.084 < x \leqslant 8.592\)	\(8.592 < x \leqslant 9.100\)	\(x > 9.100\)
   Frequency	8	22	11	9

2. Carry out a test, at the \(\frac { 1 } { 2 } \%\) significance level, of whether a normal distribution fits the data.
3. Obtain a 99\% confidence interval for the mean of all similar parallax measurements.

3 The masses, in kilograms, of a random sample of 100 chickens on sale in a large supermarket were recorded as follows.

1. Assuming that the first and last classes are the same width as the other classes, calculate an estimate of the sample mean and show that the corresponding estimate of the sample standard deviation is 0.2227 kg . A Normal distribution using the mean and standard deviation found in part (i) is to be fitted to these data. The expected frequencies for the classes are as follows.

   Mass \(( m \mathrm {~kg} )\)	\(m < 1.6\)	\(1.6 \leqslant m < 1.8\)	\(1.8 \leqslant m < 2.0\)	\(2.0 \leqslant m < 2.2\)	\(2.2 \leqslant m < 2.4\)	\(2.4 \leqslant m < 2.6\)	\(2.6 \leqslant m\)
   Expected frequency	2.17	10.92	\(f\)	33.85	19.22	5.13	0.68

2. Use the Normal distribution to find \(f\).
3. Carry out a goodness of fit test of this Normal model using a significance level of 5\%.
4. Discuss the outcome of the test with reference to the contributions to the test statistic and to the possibility of other significance levels.

6 A practice examination paper is taken by 500 candidates, and the organiser wishes to know what continuous distribution could be used to model the actual time, \(X\) minutes, taken by candidates to complete the paper. The organiser starts by carrying out a goodness-of-fit test for the distribution \(\mathrm { N } \left( 100,15 ^ { 2 } \right)\) at the \(5 \%\) significance level. The grouped data and the results of some of the calculations are shown in the following table.

1. State suitable hypotheses for the test.
2. Show how the figures 123.754 and 0.222 in the column for \(100 \leqslant X < 110\) were obtained. [3]
3. Carry out the test. The organiser now wants to suggest an improved model for the data.
4. 1. Suggest an aspect of the data that the organiser should take into account in considering an improved model.
   2. The graph of the probability density function for the distribution \(\mathrm { N } \left( 100,15 ^ { 2 } \right)\) is shown in the diagram in the Printed Answer Booklet. On the same diagram sketch the probability density function of an improved model that takes into account the aspect of the data in part (d)(i).

5. Chrystal is studying the lengths of pine cones that have fallen from a tree. She believes that the length, \(X \mathrm {~cm}\), of the pine cones can be modelled by a normal distribution with mean 6 cm and standard deviation 0.75 cm . She collects a random sample of 80 pine cones and their lengths are recorded in the table below.

1. Stating your hypotheses clearly and using a \(10 \%\) level of significance, test Chrystal's belief. Show your working clearly and state the expected frequencies, the test statistic and the critical value used.
   (10) Chrystal's friend David asked for more information about the lengths of the 80 pine cones. Chrystal told him that $$\sum x = 464 \quad \text { and } \quad \sum x ^ { 2 } = 2722.59$$
2. Calculate unbiased estimates of the mean and variance of the lengths of the pine cones. David used the calculations from part (b) to test whether or not the lengths of the pine cones are normally distributed using Chrystal's sample. His test statistic was 3.50 (to 3 significant figures) and he did not pool any classes.
3. Using a \(10 \%\) level of significance, complete David's test stating the critical value and the degrees of freedom used.
4. Estimate, to 2 significant figures, the proportion of pine cones from the tree that are longer than 7 cm . \includegraphics[max width=\textwidth, alt={}, center]{ba3f3f9c-53d2-4e95-b2f3-3f617f1821ed-15_2255_50_314_34}

1. A farmer sells strawberries in baskets. The contents of each of 100 randomly selected baskets were weighed and the results, given to the nearest gram, are shown below.

The farmer proposes that the weight of strawberries per basket, in grams, should be modelled by a normal distribution with a mean of 310 g and standard deviation 4 g . Using his model, the farmer obtains the following expected frequencies.

1. Find the value of \(a\) and the value of \(b\). Give your answers correct to one decimal place. Before \(s \leqslant 303.5\) and \(s > 315.5\) are included, for the remaining cells, $$\sum \frac { ( O - E ) ^ { 2 } } { E } = 9.71$$
2. Using a 5\% significance level, test whether the data are consistent with the model. You should state your hypotheses, the test statistic and the critical value used. An alternative model uses estimates for the population mean and standard deviation from the data given. Using these estimated values no expected frequency is below 5
   Another test is to be carried out, using a \(5 \%\) significance level, to assess whether the data are consistent with this alternative model.
3. State the effect, if any, on the critical value for this test. Give a reason for your answer.

4. Customers at a post office are timed to see how long they wait until being served at the counter. A random sample of 50 customers is chosen and their waiting times, \(x\) minutes, are summarised in Table 1. \begin{table}[h] \end{table}

1. Show that an estimate of \(\bar { x } = 5.49\) and an estimate of \(s _ { x } ^ { 2 } = 6.88\) The post office manager believes that the customers' waiting times can be modelled by a normal distribution.
   Assuming the data is normally distributed, she calculates the expected frequencies for these data and some of these frequencies are shown in Table 2. \begin{table}[h]

   Waiting Time	\(x < 3\)	\(3 - 5\)	\(5 - 6\)	\(6 - 8\)	\(x > 8\)
   Expected Frequency	8.56	12.73	7.56	\(a\)	\(b\)

   \captionsetup{labelformat=empty} \caption{Table 2}
   \end{table}
2. Find the value of \(a\) and the value of \(b\).
3. Test, at the \(5 \%\) level of significance, the manager's belief. State your hypotheses clearly.

A shoe manufacturer sees a report from another country stating that the length of adult male feet is normally distributed with a mean of 22.4 cm and a standard deviation of 2.8 cm. The manufacturer wishes to see if this model is appropriate for his customers and collects data on the length, correct to the nearest cm, of the right foot of a random sample of 200 males giving the following results: The expected frequencies for the \(\leq 18\) and \(19 - 21\) groups are calculated as 16.46 and 58.44 respectively, correct to 2 decimal places.

1. Calculate expected frequencies for the other three classes. [7]
2. Stating your hypotheses clearly, test at the 10\% level of significance whether or not this data can be modelled by the distribution N(22.4, 2.8²). [7]

The manufacturer wishes to refine the model by not assuming a mean and standard deviation.

1. Explain briefly how the manufacturer should proceed. [3]