Question

The number of carbohydrates found in a random sample of fast - food entrees is listed below. Is there sufficient evidence to conclude that the variance differs from 100? Use the 0.05 level of significance. Assume the variables are approximately normally distributed.

44    51    47     38     43

39    53    40    62     46

73    61     63    53     39  

52    30    41    38

a) State the hypotheses and identify the claim with the correct hypothesis.

Answer 1

$$\bar{x}=\dfrac{\sum_ix_i}{n}=\dfrac{913}{19}\approx48.0526$$

$$Var(x)=s^2=\dfrac{\sum_i(x_i-\bar{x})^2}{n-1}\approx118.608187$$

$$s=\sqrt{s^2}\approx10.8907$$

a) The following null and alternative hypotheses need to be tested:

$H_0: \sigma^2=100$

$H_1: \sigma^2\not=100$

​This corresponds to a two-tailed test, for which a Chi-Square test for one population variance will be used.

Based on the information provided, the significance level is $\alpha=0.05,$ and the the rejection region for this two-tailed test is $R=\{χ ^2:χ^2<8.231$ or $χ^2>31.526\}.$