Goodness of fit when parameters are estimated Problem set A

A bakery records the number of burnt loaves in each of 50 batches. 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 6 groups. State the degrees of freedom.
2. (b) A binomial distribution B(7, 0.3) is fitted to data. The value 0.3 comes from a national survey, not from this data. After combining groups there are 7 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 8 groups. State the degrees of freedom.
4. (d) A company says the mean number of faults per day is 2.5. One student tests whether 40 days of data fit Po(2.5). Another student tests whether the data fit a Poisson distribution with the mean taken from the 40 days. Which student has a parameter that is estimated? How do their degrees of freedom compare, for the same number of groups?

The journey times, in minutes, of 80 commuters 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 marks of 100 candidates are to be compared with a normal distribution with mean 52 and standard deviation 8. The marks are put into six groups: under 36, 36 to under 44, 44 to under 52, 52 to under 60, 60 to under 68, 68 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.

A call centre records the number of dropped calls in each of 100 hours. 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 museum says it receives a mean of 1.8 school group visits per day. The number of visits on each of 80 days is recorded. Test, at the 5% significance level, whether Po(1.8) 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.

100 students each answer 5 quiz questions and the number of correct answers is recorded. The teacher says each answer is correct with probability 0.7. Use a 5% significance level.

1. (a) Test whether B(5, 0.7) 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 test scores of 100 students are grouped as shown. The teacher says the mean score is 60. 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 heights of 180 sunflower plants are recorded and grouped as shown. A student thinks the heights 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 heights follow a normal distribution, and state your conclusion.

A game spinner has five sectors numbered 1 to 5. The maker says the number X it shows has probability function \(P(X=x)=\dfrac{x}{15}\). The spinner is spun 75 times. The numbers 1 to 5 occur 9, 10, 16, 20, 20 times. A teacher picks a student by number from 1 to 5, 80 times in all. The numbers 1 to 5 are picked 10, 20, 25, 15, 10 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 scientist breeds fruit flies and expects long and short wings in the ratio 3 : 1. Of 570 flies, 416 have long wings and 154 have short wings. Test, at the 5% significance level, whether the data agree with the ratio.

A survey asks 227 shoppers how they usually travel to the shops and on which day of the week they shop. Test, at the 5% significance level, whether how a shopper travels is independent of the day of the week. Combine categories if you need to.

The number of customers arriving at a till in each of 100 one-minute periods 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 spins a spinner with 5 equal-looking sectors 60 times. The results are shown.

1. (a) Test, at the 5% significance level, whether the spinner 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 2.3, 6.8, 11.4 and 4.1. Explain why the test should not be used with these values, and say what to do.
2. (b) A school asks every pupil in the canteen queue: 'Do you agree that the disappointing canteen food should be improved?' Comment on the validity of this questionnaire.