5.06b Fit prescribed distribution: chi-squared test

Quality control inspectors in a factory are investigating the lengths of glass tubes that will be used to make laboratory equipment.

1. Data on the observed lengths of a random sample of 200 glass tubes from one batch are available in the form of a frequency distribution as follows.

   Length \(x\) (mm)	Observed frequency
   \(x \leq 298\)	1
   \(298 < x \leq 300\)	30
   \(300 < x \leq 301\)	62
   \(301 < x \leq 302\)	70
   \(302 < x \leq 304\)	34
   \(x > 304\)	3

   The sample mean and standard deviation are 301.08 and 1.2655 respectively. The corresponding expected frequencies for the Normal distribution with parameters estimated by the sample statistics are

   Length \(x\) (mm)	Expected frequency
   \(x \leq 298\)	1.49
   \(298 < x \leq 300\)	37.85
   \(300 < x \leq 301\)	55.62
   \(301 < x \leq 302\)	58.32
   \(302 < x \leq 304\)	44.62
   \(x > 304\)	2.10

   Examine the goodness of fit of a Normal distribution, using a 5\% significance level. [7]
2. It is thought that the lengths of tubes in another batch have an underlying distribution similar to that for the batch in part (i) but possibly with different location and dispersion parameters. A random sample of 10 tubes from this batch gives the following lengths (in mm). 301.3 \quad 301.4 \quad 299.6 \quad 302.2 \quad 300.3 \quad 303.2 \quad 302.6 \quad 301.8 \quad 300.9 \quad 300.8
   1. Discuss briefly whether it would be appropriate to use a \(t\) test to examine a hypothesis about the population mean length for this batch. [2]
   2. Use a Wilcoxon test to examine at the 10\% significance level whether the population median length for this batch is 301 mm. [9]

Commentators on a game of cricket say that a certain batsman is "playing shots all round the ground". A sports statistician wishes to analyse this claim and records the direction of shots played by the batsman during the course of his innings. She divides the \(360°\) around the batsman into six sectors, measuring the angle of each shot clockwise from the line between the wickets, and obtains the following results: Stating your hypotheses clearly and using a 5% level of significance test whether or not these data can be modelled by a continuous uniform distribution. [9]

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]

The discrete random variable \(X\) is equally likely to take values 0, 1 and 2. \(3N\) observations of \(X\) are obtained, and the observed frequencies corresponding to \(X = 0\), \(X = 1\) and \(X = 2\) are given in the following table.

\(x\)	0	1	2
Observed frequency	\(N - 1\)	\(N - 1\)	\(N + 2\)

The test statistic for a chi-squared goodness of fit test for the data is 0.3. Find the value of \(N\). [4]

A biased spinner has five sides, numbered 1 to 5. Elmer spins the spinner repeatedly and counts the number of spins, \(X\), up to and including the first time that the number 2 appears. He carries out this experiment 100 times and records the frequency \(f\) with which each value of \(X\) is obtained. His results are shown in Table 1, together with the values of \(xf\).

\(x\)	1	2	3	4	5	6	\(\geqslant 7\)	Total
Frequency \(f\)	20	15	9	13	10	10	23	100
\(xf\)	20	30	27	52	50	60	161	400

Table 1

1. State an appropriate distribution with which to model \(X\), determining the value(s) of any parameter(s). [3]

Elmer carries out a goodness-of-fit test, at the 5\% level, for the distribution in part (a). Table 2 gives some of his calculations, in which numbers that are not exact have been rounded to 3 decimal places.

\(x\)	1	2	3	4	5	6	\(\geqslant 7\)
Observed frequency \(O\)	20	15	9	13	10	10	23
Expected frequency \(E\)	25	18.75	14.063	10.547	7.910	5.933	17.798
\((O - E)^2/E\)	1	0.75	1.823	0.571	0.552	2.789	1.520

Table 2

1. Show how the expected frequency corresponding to \(x \geqslant 7\) was obtained. [2]
2. Carry out the test. [5]

At a bird feeding station, birds are captured and ringed. If a bird is recaptured, the ring enables it to be identified. The table below shows the number of recaptures, \(x\), during a period of a month, for each bird of a particular species in a random sample of \(40\) birds.

Number of recaptures, \(x\)	0	1	2	3	4	5	6	7	8	9	10
Frequency	2	5	5	9	10	4	3	1	0	1	0

1. The sample mean of \(x\) is \(3.4\). Calculate the sample variance of \(x\). [2]
2. Briefly comment on whether the results of part (i) support a suggestion that a Poisson model might be a good fit to the data. [1]

The screenshot below shows part of a spreadsheet for a \(\chi^2\) test to assess the goodness of fit of a Poisson model. The sample mean of \(3.4\) has been used as an estimate of the Poisson parameter. Some values in the spreadsheet have been deliberately omitted. \includegraphics{figure_2}

1. State the null and alternative hypotheses for the test. [1]
2. Calculate the missing values in cells
3. Complete the test at the \(10\%\) significance level. [5]
4. The screenshot below shows part of a spreadsheet for a \(\chi^2\) test for a different species of bird. Find the value of the Poisson parameter used. \includegraphics{figure_3} [3]

A life insurance saleswoman investigates the number of policies she sells per day. The results for a random sample of 50 days are shown in the table below.

Number of policies sold	0	1	2	3	4	5	6
Number of days	2	2	9	12	15	9	1

She sees the same fixed number of clients each day. She would like to know whether the binomial distribution with parameters 6 and 0·6 is a suitable model for the number of policies she sells per day.

1. State suitable hypotheses for a goodness of fit test. [1]
2. Here is part of the table for a \(\chi^2\) goodness of fit test on the data.

   Number of policies sold	0	1	2	3	4	5	6
   Observed	2	2	9	12	15	9	1
   Expected	0·205	1·843	6·912	\(d\)	\(e\)	9·331	2·333

   1. Calculate the values of \(d\) and \(e\).
   2. Carry out the test using a 10% level of significance and draw a conclusion in context. [10]
3. What do the parameters 6 and 0·6 mean in this context? [1]

A company has 20 boats to hire out. Payment is always taken in advance and all 20 boats are hired out each day. A manager at the company notices that 10% of groups do not turn up to take the boats, despite having already paid to hire them. The manager wishes to investigate whether the numbers of boats that do not get taken each day can be modelled by the binomial distribution B\((20, 0 \cdot 1)\). The numbers of boats that were not taken for 110 randomly selected days are given below.

Number of boats not taken	0	1	2	3	4	5 or more
Frequency	10	35	29	25	8	3

1. State suitable hypotheses to carry out a goodness of fit test. [1]
2. Here is part of the table for a \(\chi^2\) goodness of fit test on the data.

   Number of boats not taken	0	1	2	3	4	5 or more
   Observed	10	35	29	25	8	3
   Expected	\(f\)	29·72	\(g\)	20·91	9·88	4·75

   1. Calculate the values of \(f\) and \(g\).
   2. By completing the test, give the conclusion the manager should reach. [10]

The cost of hiring a boat is £15. Since demand is high and the proportion of groups that do not turn up is also relatively high, the manager decides to take payment for 22 boats each day. She would give £20 (a full refund and some compensation) to any group that has paid and turned up, but cannot take a boat out due to the overselling. Assume that the proportion of groups not turning up stays the same.

1. 1. Suggest a binomial model that the manager could use for the number of groups arriving expecting to hire a boat.
   2. Hence calculate the expected daily net income for the company following the manager's decision. [8]
2. Is the manager justified in her decision? Give a reason for your answer. [1]

The manager of a hockey team studies last season's results and puts forward the theory that the number of goals scored per match by her team can be modelled by a Poisson distribution with mean 2.0. The number of goals scored during the season are summarised below.

Goals scored	0	1	2	3	4 or more
Frequency	6	11	15	10	8

1. State suitable hypotheses to carry out a goodness of fit test. [1]
2. Carry out a \(\chi^2\) goodness of fit test on this data set, using a 5% level of significance and draw a conclusion in context. [9]

The discrete random variable \(X\) is equally likely to take values 0, 1 and 2. \(3N\) 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 \(N\). [4]