Question

41 Q3.2. (based on Problem 10.20; page 356 & 10,47 & 10.51; page 359) A random sample of 64 bags of white cheddar popcorn weighed, on average, 5.44 ounces with a standard deviation of 0.24 ounce. a. Test the hypothesis that u = 5.55 ounces against the alternative hypothesis, 4 < 5.55 ounces, at the 0.05 level of significance. Comment b. How large a sample is required if we want the power of the test to be 0.90 when: i. the true mean is 5.35? Assume that o = 0.24. ii. the true mean exceeds the hypothesized value by 1.40.

Answer 1

a)

Η :μ = 5.55

H, :μ< 5.55

Standard error of mean, SE = s / $\sqrt{n}$ = 0.24 / $\sqrt{64}$ = 0.03

Test statistic, t = ($\bar{x}$ - $\mu$ ) / SE = (5.44 - 5.55) / 0.03 = -3.67

Degree of freedom, df = n-1 = 64-1 = 63

P-value = P(t < -3.67) = 0.00025

Since, p-value is less than 0.05 significance level, we reject null hypothesis H0 and conclude that there is strong evidence that population mean weight of cheddar popcorn is less than 5.55 ounces.

b.

i.

Standard error of mean, SE = $\sigma$ / $\sqrt{n}$ = 0.24 / $\sqrt{n}$

Critical value of z at 0.05 significance level for left tail test is 1.645

Critical value of sample mean to reject H0 = 5.55 - 1.645 * 0.24 / $\sqrt{n}$ = 5.55 - 0.3948 / $\sqrt{n}$

Power = P(Reject H0 | $\mu$ = 5.35) = 0.90

P($\bar{x}$ < 5.55 - 0.3948 / $\sqrt{n}$ | $\mu$ = 5.35) = 0.90

P[Z < (5.55 - 0.3948 / $\sqrt{n}$ - 5.35 )/(0.24 / $\sqrt{n}$ )] = 0.90

(0.2 - 0.3948 / $\sqrt{n}$ ) = 1.28 * 0.24 / $\sqrt{n}$

0.2 - 0.3948 / $\sqrt{n}$ = 0.3072 / $\sqrt{n}$

0.702/$\sqrt{n}$ = 0.2

n = (0.702 / 0.2)2  = 13 (Rounding to next integer)

ii.

True mean = 5.55 - 1.4$\sigma$ = 5.55 - 1.4 * 0.24 = 5.214

Critical value of sample mean to reject H0 = 5.55 - 1.645 * 0.24 / $\sqrt{n}$ = 5.55 - 0.3948 / $\sqrt{n}$

Power = P(Reject H0 | $\mu$ = 5.214) = 0.90

P($\bar{x}$ < 5.55 - 0.3948 / $\sqrt{n}$ | $\mu$ = 5.214) = 0.90

P[Z < (5.55 - 0.3948 / $\sqrt{n}$ - 5.214​​​​​​​ )/(0.24 / $\sqrt{n}$ )] = 0.90

(0.336- 0.3948 / $\sqrt{n}$ ) = 1.28 * 0.24 / $\sqrt{n}$

0.336 - 0.3948 / $\sqrt{n}$ = 0.3072 / $\sqrt{n}$

0.702/$\sqrt{n}$ = 0.336

n = (0.702 / 0.336)2  = 5 (Rounding to next integer)