Question

5. A U.S dime weighs 2.260 grams when minted. A random sample of 20 circulated dimes showed a mean weight of 2.248 grams with a standard deviation of .024 grams. Using alpha = 0.01, is the mean weight of all circulated dimes lower than the mean weight? What is the Excel formula to calculate the p-value?

Answer 1

Aim: To test whether the mean weight of all circulated dimes lower than the mean weight 2.260 grams.

Null hypothesis H₀: The mean weight of all circulated dimes are greater than or equal to the mean weight 2.260.

Alternate hypothesis H_a: The mean weight of all circulated dimes are lesser than the mean weight 2.260..

Appropriate test: The sample size is lesser than 30 and the population standard deviation is not known. So students' t-test for one sample mean is the appropriate test to be used.

Test statistic:

The test statistic for the test is given as and the sampling distribution is t_n-1 where (n - 1) is the degrees of freedom.

Observed test statistic:

Get the observed test statistic by substituting the respective values in the test statistic.

Decision Rule:

The level of significance is 0.01 and the test is left tailed test as the alternate hypothesis states the rejection area in the left tail. The decision rule is to reject the null hypothesis if either the observed test statistic is lesser than the left critical value of the t₁₉ distribution at 1% level of significance or the p-value of the observed test statistic is lesser than the level of significance.

Critical value & Decision rule:

The left critical value of t₁₉ distribution at 1% level of significance is -2.539 as from the t-table. The p-value of the observed test statistic is given by the probability P(t₁₉ < -2.539).

= P(t₁₉ > 2.539)

= 0.01877

P-value is computed using the Excel formula tdist(2.539, 19, 1)

Conclusion:

The observed test statistic is greater than the left critical value and also the p-value is greater than the level of significance. According to the decision rule, there is no evidence to reject the null hypothesis. We fail to reject the null hypothesis based on the evidence of the sample provided at 1% level of significance. So it is reasonable to conclude that the mean of all circulated dimes are greater than or equal to the overall mean 2.260 grams and the deviation noticed in the sample mean is by chance only.