Testing the fit of a model Problem set A

The capacity of a type of battery, in mAh, is claimed to follow \(N(500, 40^2)\). For a sample of 300 batteries the capacities are grouped as: below 440, 440 to 500, 500 to 540, and 540 or more. Find the expected frequency of each group.

A quiz has 3 questions. Each is answered correctly with probability 0.4, independently. The quiz is taken by 250 students. Let X be the number of correct answers for one student.

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

A teacher models the number of mistakes X (from 1 to 4) that a student makes in a quiz by \(P(X = x) = \frac{x^2}{30}\). There are 240 students. Find the expected frequency of each number of mistakes.

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 heights of trees follow \(N(20, 3^2)\).
2. (b) Testing whether the number of faulty bulbs in a box of 5 follows \(B(5, 0.3)\).
3. (c) A table with 6 groups, no values estimated from the data.

A goodness of fit test has observed frequencies 26, 55, 104, 77, 38 and expected frequencies 20, 60, 100, 80, 40.

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

In the quiz above, the numbers of students with 0, 1, 2, 3 correct answers were 66, 92, 76 and 16. Test at the 5% level whether the number of correct answers follows \(B(3, 0.4)\). 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 journey times, in minutes, of 200 workers are grouped as below 28, 28 to 32, 32 to 36, 36 to 40, and 40 or more, with frequencies 30, 58, 52, 38, 22. Test at the 5% level whether the times follow \(N(34, 5^2)\).

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 six-sided die is rolled 150 times. The faces 1 to 6 come up 20, 31, 22, 30, 18 and 29 times. Test at the 2% significance level whether the die 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.

Five coins are thrown together and the number of heads X is recorded. This is done 320 times. The frequencies of 0, 1, 2, 3, 4, 5 heads are 4, 28, 86, 121, 67 and 14. Test at the 5% level whether the coins are fair.

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 librarian models the number of books X (from 1 to 5) borrowed in one visit by \(P(X = x) = \frac{6 - x}{15}\). On 300 visits the numbers of books borrowed were 1 to 5: 90, 95, 52, 38 and 25. 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.