Question

The wind speed in Laramie, Wyoming is normally distributed. The average wind speed for the state of Wyoming is approximately 20 mph. A curious student wanted to find out if the wind speed in Laramie was different than the state average. They took a sample of 25 random days in Laramie, measured the wind speed, and found an average of 18 mph with a standard deviation of 7.8 miles per hour.

Test the claim that the average wind speed in Laramie is less than the state average. Provide all six steps to the hypothesis test, using both a critical value AND a p-value to make your decision about the null.

Provide the confidence interval estimate of the true wind speed in Laramie.

Explain how your hypothesis test in 8_1 and your confidence interval in 8_2 are consistent.

Answer 1

Let be the mean wind speed in Laramie.

Step 1:

Null Hypothesis H0:  = 20 mph

Altenative Hypothesis H0: < 20 mph

Step 2:

Since we do not know the true population standard deviation we will conduct one sample t test.

Step 3:

Significance level, = 0.05

Degree of freedom = n-1 = 25-1 = 24

Critical value of t at = 0.05 and df = 24 is -1.71

We reject the null hypothesis if the test statistic is less than -1.71 or p-value is less than 0.05.

Step 4:

Sample mean, = 18

Sample standard deviation s = 7.8

Standard error of mean, SE = s /  = 7.8 /  = 1.56

Test statistic, t = ( - ) / SE =  (18 - 20) / 1.56 = -1.282

p-value = p(t < -1.282, df = 24) =  0.1060

Step 5:

Since, p-value is greater than 0.05 significance level, we fail to reject null hypothesis H0.

Since the test statistic, t is greater than the criticak value and does not lie in the rejection region, we fail to reject null hypothesis H0.

Step 6:

We conclude that there is no significant evidence that true mean wind speed in Laramie is less than the state average of 20 mph.

Confidence interval estimate of the true wind speed in Laramie

Z value for 95% confidence interval is 1.96

95% confidence interval of the true wind speed in Laramie is,

(18 - 1.96 * 1.56 ,  18 + 1.96 * 1.56)

(14.94 , 21.06)

Since the confidence interval estimate of the true wind speed in Laramie contains the values greater than 20 mph, we are 95% confident that there is no significant evidence that true mean wind speed in Laramie is less than the state average of 20 mph.

Thus, hypothesis test in and confidence interval in are consistent.