Answer to Question #154649 in Statistics and Probability for chi 2

Question

Grades assigned by an economics instructor

have historically followed a symmetrical distribution:

5% A’s, 25% B’s, 40% C’s, 25% D’s, and

5% F’s. This year, a sample of 150 grades was drawn

and the grades (1 = A, 2 = B, 3 = C, 4 = D, and

5 = F) were recorded. Can you conclude, at the

10% level of significance, that this year’s grades are

distributed differently from grades in the past?

Answer 1

Grades are assigned by an economics instructor have historically followed a symmetrical distribution.

The claim is that the grades are distributed differently from the grades in the past.

From the given data we have, the null and alternative hypotheses are

H₀: p₁ = 0.05, p₂ = 0.25, p₃ = 0.40, p₄ = 0.25, p₅ = 0.05

H₁: At least one p_i is not equil to its specific value

The test statistic for the Chi-Square goodness-of-fit (χ²) is given by,

$χ^2 = \sum^{k}_{i=1}\frac{(f_i-e_i)^2}{e_i}$

e_i = expected frequencies

f_i = observed frequencies

Given level of significance α=0.10

Using MINITAB, the following are the steps to get the χ²-statistic for the Account receivable data.

1) Click Stat, Table, and Chi-square Goodness-of-fit Test (One Variable)….

2) Type the observed values into the Observe Counts and specify the variable name.

3) Click Proportions specified by historical counts and Input constants type the values of the proportions under the null hypothesis (0.05, 0.25, 0.40, 0.25, 0.05)

4) Click OK.

We get the following output

From the above output,

The test statistic χ² = 14.0667

The P-value is 0.007

Here we observe that, the P-value is less than the level of the significance 0.10, so we reject the null hypothesis. Therefore, there is enough evidence to support the claim that the grades are distributed differently from the grades in the past.