Goodness of fit when parameters are estimated Problem set B

A bike shop records the number of punctures it repairs for each of 50 customers. The results are shown. A Poisson model is to be tested.

1. (a) Use the calculator to find the mean of the data.
2. (b) Write down the Poisson distribution that uses this mean and find P(X = 2) under it.

Find the number of degrees of freedom for each goodness of fit test, and answer the last part in words.

1. (a) A Poisson distribution is fitted to data and its mean is found from the data. After combining groups there are 7 groups. State the degrees of freedom.
2. (b) A binomial distribution B(10, 0.45) is fitted to data. The value 0.45 is stated by the manufacturer, not found from this data. After combining groups there are 5 groups. State the degrees of freedom.
3. (c) A normal distribution is fitted to data. Both the mean and the standard deviation are calculated from the data. After combining groups there are 9 groups. State the degrees of freedom.
4. (d) A coach says a player scores each penalty with probability 0.6. One analyst tests whether the number scored in 5 penalties, over 50 rounds, fits B(5, 0.6). Another analyst tests whether the data fit B(5, p) with p found from the 50 rounds. Which analyst has a parameter that is estimated? How do their degrees of freedom compare, for the same number of groups?

The heights, in centimetres, of 60 students are recorded. The data are grouped as shown.

1. (a) Write down the midpoint of each class and use them to estimate the mean.
2. (b) Estimate the standard deviation.

The lengths of 150 fish are to be compared with a normal distribution with mean 40 cm and standard deviation 6 cm. The lengths are put into seven groups: under 30 cm, 30 to under 35 cm, 35 to under 40 cm, 40 to under 45 cm, 45 to under 50 cm, 50 to under 56 cm, 56 cm or over.

1. (a) Find the expected frequencies for the groups, using the open-ended first and last groups.
2. (b) Which groups must be combined so that every expected frequency is greater than 5? Write down the new groups and the number of groups.

The number of goals scored in each of 100 football matches is recorded. The results are shown. Test, at the 5% significance level, whether a Poisson distribution is a suitable model.

1. (a) Find the mean and write down the Poisson distribution that will be tested.
2. (b) State the hypotheses, find the expected frequencies and combine groups so that every expected frequency is greater than 5.
3. (c) Find the degrees of freedom, χ² and the p-value, and state your conclusion.

A shop says it receives a mean of 3.4 online orders per hour. The number of orders in each of 60 hours is recorded. Test, at the 5% significance level, whether Po(3.4) is a suitable model.

1. (a) State the hypotheses.
2. (b) Find the expected frequencies, say which groups must be combined, and state the degrees of freedom.
3. (c) Find the p-value and state your conclusion.

A player takes 10 free throws in each of 100 practice sessions and the number scored is recorded. The coach says the player scores each throw with probability 0.5. Use a 5% significance level.

1. (a) Test whether B(10, 0.5) is a suitable model.
2. (b) Find an estimate of p from the data.
3. (c) Test whether a binomial distribution, with p estimated in part (b), is a suitable model.

The delivery times, in minutes, of 90 orders are grouped as shown. The company says the mean delivery time is 44 minutes. Treat the first and last classes as open-ended. Use a 5% significance level.

1. (a) Use the class midpoints to estimate the standard deviation.
2. (b) Find the expected frequencies, combine groups so that every expected frequency is greater than 5, and state the degrees of freedom.
3. (c) Test whether the data follow a normal distribution with this mean, and state your conclusion.

The times taken by 120 runners to complete a 5 km race are recorded and grouped as shown. A club coach thinks the times follow a normal distribution. Use a 5% significance level.

1. (a) Use the class midpoints to estimate the mean and the standard deviation.
2. (b) Find the expected frequencies for a normal distribution with these values, treating the first and last classes as open-ended. Combine groups so that every expected frequency is greater than 5, and state the degrees of freedom.
3. (c) Test whether the times follow a normal distribution, and state your conclusion.

A shop says the number of scoops X that a customer buys has probability function \(P(X=x)=\dfrac{5-x}{10}\) for \(x=1,2,3,4\). The scoops bought by 80 customers are recorded. The numbers 1 to 4 occur 25, 25, 20, 10 times. A spinner has six equal-looking sectors numbered 1 to 6. It is spun 90 times. The numbers 1 to 6 occur 9, 21, 14, 14, 18, 14 times.

1. (a) Test, at the 5% significance level, whether the model suggested in the first paragraph is suitable.
2. (b) Test, at the 5% significance level, whether the second situation is modelled by a uniform distribution.

A gardener crosses red and white snapdragons and expects red, pink and white flowers in the ratio 1 : 2 : 1. Of 200 plants, 47 are red, 96 are pink and 57 are white. Test, at the 5% significance level, whether the data agree with the ratio.

226 people using a library are asked how they travel there. They are grouped by age. Test, at the 5% significance level, whether how a person travels to the library is independent of age group. Combine categories if you need to.

The number of insects found on each of 100 plants is recorded. The results are shown.

1. (a) Find an unbiased estimate of the mean.
2. (b) Find an unbiased estimate of the variance.
3. (c) Compare your answers and say whether a Poisson distribution is a plausible model. Give a reason.

A student rolls a 6-sided die 60 times. The results are shown.

1. (a) Test, at the 5% significance level, whether the die is fair.
2. (b) Every frequency is now multiplied by 5, so the proportions are the same. Test again at the 5% significance level.
3. (c) Explain why the conclusions differ.

Answer both parts in words.

1. (a) A goodness of fit test has expected frequencies 1.8, 4.2, 9.5 and 12.5. Explain why the test should not be used with these values, and say what to do.
2. (b) A school asks only the members of its gym club how many hours of exercise pupils take each week. Comment on the validity of this questionnaire.