Testing the fit of a model Problem set B

A commuter's journey time, in minutes, is claimed to follow \(N(25, 4^2)\). For 250 journeys the times are grouped as: below 19, 19 to 25, 25 to 29, and 29 or more. Find the expected frequency of each group.

A footballer takes 3 penalties in a match. Each scores with probability 0.7, independently. She plays 200 such matches. Let X be the number of penalties scored in one match.

1. (a) State the distribution of X.
2. (b) Find the expected frequency of each value of X.

A shop models the number of items X (from 1 to 4) in a customer's basket by \(P(X = x) = \frac{x+2}{18}\). There are 270 customers. Find the expected frequency of each number of items.

Write \(H_0\) and \(H_1\) for each goodness of fit test, and state the degrees of freedom for (c).

1. (a) Testing whether the masses of apples follow \(N(150, 12^2)\).
2. (b) Testing whether the number of defective chips in a pack of 6 follows \(B(6, 0.2)\).
3. (c) A table with 7 groups, no values estimated from the data.

A goodness of fit test has observed frequencies 11, 58, 92, 66, 23 and expected frequencies 15, 50, 100, 60, 25.

1. (a) State the degrees of freedom.
2. (b) Use the calculator to find \(\chi^2\) and the p-value.

In the matches above, the numbers with 0, 1, 2, 3 penalties scored were 8, 31, 95 and 66. Test at the 5% level whether the number of penalties scored follows \(B(3, 0.7)\). The critical value of \(\chi^2\) with df = 3 at the 5% level is 7.815.

1. (a) Write \(H_0\) and \(H_1\) and find the expected frequencies.
2. (b) Find \(\chi^2\) on the calculator.
3. (c) Compare \(\chi^2\) with the critical value and write the conclusion in context.

The reaction times, in milliseconds, of 250 players are grouped as below 102, 102 to 114, 114 to 126, 126 to 138, and 138 or more, with frequencies 36, 70, 64, 52, 28. Test at the 5% level whether the times follow \(N(120, 15^2)\).

1. (a) Find the expected frequencies.
2. (b) Write \(H_0\) and \(H_1\), and state the degrees of freedom.
3. (c) Find \(\chi^2\) and the p-value from the calculator.
4. (d) Write the conclusion in context.

A wheel at a school fair has six equal sectors numbered 1 to 6. It is spun 210 times. The numbers 1 to 6 come up 30, 41, 33, 39, 28 and 39 times. Test at the 2% significance level whether the wheel is fair.

1. (a) Write \(H_0\) and \(H_1\).
2. (b) State the expected frequency and the degrees of freedom.
3. (c) Find \(\chi^2\) and the p-value from the calculator.
4. (d) Write the conclusion in context.

A gardener plants 4 seeds in each of 320 pots. Each seed is thought to germinate with probability 0.5, independently. The numbers of pots with 0, 1, 2, 3, 4 seeds germinating are 12, 62, 118, 94 and 34. Test at the 5% level whether the model for each seed is correct.

1. (a) State the distribution of X if \(H_0\) is true, and write \(H_0\) and \(H_1\).
2. (b) Find the expected frequencies.
3. (c) State the degrees of freedom and find \(\chi^2\) and the p-value on the calculator.
4. (d) Write the conclusion in context.

A café owner models the number of hot drinks X (from 1 to 5) ordered by a table by \(P(X = x) = \frac{x + 1}{20}\). At 240 tables the numbers of hot drinks ordered were 1 to 5: 30, 32, 52, 54 and 72. Test at the 5% level whether the model fits.

1. (a) Find the expected frequencies.
2. (b) Write \(H_0\) and \(H_1\), and state the degrees of freedom.
3. (c) Find \(\chi^2\) and the p-value on the calculator and write the conclusion in context.