Question

The Student's t distribution table gives critical values for the Student's t distribution. Use an appropriate d.f. as the row header. For a right-tailed test, the column header is the value of α found in the one-tail area row. For a left-tailed test, the column header is the value of α found in the one-tail area row, but you must change the sign of the critical value t to −t. For a two-tailed test, the column header is the value of α from the two-tail area row. The critical values are the ±t values shown.

Pyramid Lake is on the Paiute Indian Reservation in Nevada. The lake is famous for cutthroat trout. Suppose a friend tells you that the average length of trout caught in Pyramid Lake is μ = 19 inches. However, a survey reported that of a random sample of 51 fish caught, the mean length was x = 18.4 inches, with estimated standard deviation s = 2.8 inches. Do these data indicate that the average length of a trout caught in Pyramid Lake is less than μ = 19 inches? Use α = 0.05. Solve the problem using the critical region method of testing (i.e., traditional method). (Round the your answers to three decimal places.)

test statistic	=
critical value	=

Answer 1

To test against

Here

sample mean

sample standard deviation

and sample size n = 51

The test statistic can be written as

which under H₀ follows a t distribution with n-1 df.

We reject H₀ at 5% level of significance if

Now,

The value of the test statistic =

and critical value

Since , so we fail to reject H₀ at 5% level of significance and we can conclude that the average length of a trout caught in Pyramid Lake is not significantly less than 19 inches.