Question

A sign on a barrel of nuts in a supermarket says that it contains 30% cashews, 30% hazelnuts, and 40% peanuts by weight. You mix up the nuts and scoop out 18 pounds. When you weigh the nuts, you find that you have 5 pounds of cashews, 4 pounds of hazelnuts, and 9 pounds of peanuts. Is there evidence to doubt the supermarket's claim

Answer 1

The calculated χ² = 0.842 does not fall in the critical region χ² ≥ 5.99 so we accept the null hypothesis that all the proportions are equal and there is not enough evidence to doubt the supermarket's claim.

Explanation:

1) We set up our null and alternative hypothesis as

H0: p1= 5/18, p2= 4/18, p3= 9/18

against the claim

Ha: p1≠ 5, p2≠ 4, p3≠ 9

2) the significance level alpha is set at 0.05

3) the test statistic under H0 is

χ²= ∑ (O - E)²/ E where O is the observed and E is the expected frequency

which has an approximate chi square distribution with 2 d.f

4) Computations:

Observed Expected χ²= ∑ (O - E)²/ E

5 5.4 0.0296

4 5.4 0.3629

9 7.2 0.45

∑ 0.842

5) The critical region is χ² ≥ χ² (0.05)2 = 5.99

6) Conclusion:

The calculated χ² = 0.842 does not fall in the critical region χ² ≥ 5.99 so we accept the null hypothesis that all the proportions are equal and there is not enough evidence to doubt the supermarket's claim.