Spreadsheet-based chi-squared test

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.

6 A researcher is investigating whether there is any relationship between whether a cyclist wears a helmet and the distance, \(x \mathrm {~m}\), the cyclist is from the kerb (the edge of the road). Data are collected at a particular location for a random sample of 250 cyclists. The researcher carries out a chi-squared test. Fig. 6 is a screenshot showing part of a spreadsheet used to analyse the data. 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.
   Carry out a hypothesis test at the \(10 \%\) significance level to investigate whether there is any association between helmet wearing and distance from the kerb.
2. Discuss briefly what the data suggest about helmet wearing for different distances from the kerb.

3 A student wants to know whether there is any association between age and whether or not people smoke. The student takes a sample of 120 adults and asks each of them whether or not they smoke. Below is a screenshot showing part of a spreadsheet used to analyse the data. Some values in the spreadsheet have been deliberately omitted.

1. The student wants to carry out a chi-squared test to analyse the data. State a requirement of the sample if the test is to be valid. For the rest of this question, you should assume that this requirement is met.
2. Determine the missing values in each of the following cells.
   Carry out a hypothesis test at the \(5 \%\) significance level to investigate whether there is any association between age and smoking status.
3. Discuss what the data suggest about the smoking status for each different age group.

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.

9 A random sample of adults in the UK were asked to state their primary source of news: television (T), internet (I), newspapers (N) or radio (R). The responses were classified by age group, and an analysis was carried out to see if there is any association between age group and primary source of news. Fig. 9 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] \end{table}

1. (A) State the sample size.
   (B) Give the name of the appropriate hypothesis test.
   (C) State the null and alternative hypotheses.
2. Showing your calculations, find the missing values in cells

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]