Question

The weights of four randomly and independently selected bags of potatoes labeled 20 pounds were found to be 21.1​, 21.4​, 20.7​, and 21.4. Assume Normality.

a. Using a​ two-sided alternative​ hypothesis, should you be able to reject the hypothesis that the population mean is 20 pounds using a significance level of 0.05​? Why or why​ not? The confidence interval is reported​ here: I am 95​% confident the population mean is between 20.6 and 21.7 pounds.

b. Test the hypothesis that the population mean is not 20. Use a significance level of 0.05.

c. What is the test statistic and P Value?

Answer 1

a )

We have the hypothesis as

We have given the confidence interval

95% confidence interval for the population mean is 20.6 to 21.7

We have given as the population mean = 20 in the claim

When the population mean do not lie in the confidence interval then we reject H₀

Decision rule from the confidence interval .

1 ) when given   value of the population mean lies in the confidence interval then we fail to reject H₀

2 ) When the given population mean lies outside the confidence interval then we Reject H₀

Here 20 do not lie between 20.6 and 21.7 so our decision is Reject H₀

b ) and C )

n = 4 ( we have given 4 values )

We use the test statistics formula

df = degrees of freedom = n - 1 = 4 - 1 = 3

Here our test is 2 tailed because it has not equal to sign .

When alternative hypothesis (H_a) has > or < sign then test is one tailed

We use excel to find the p value

Select excel empty cell and type

=TDIST(6.934760925,3,2)

Press " enter "

Do not forget to type = sign

test statistics = 6.934760925 df = degrees of freedom = 3 tail of the test = 2

P value = 0.006148723

level of significance = 0.05

Decision rule

So we get the decision as

We fail to reject H₀ because p value = 0.006148723 is less than 0.05

(Note we get the same Every time from Confidence interval and the p value )

H₀: means the Null hypothesis .

H_a: means the alternative hypothesis .