Question

It is thought that the mean length of trout in lakes in a certain region is 20 inches. A sample of 46 trout from one particular lake had a sample mean of 18.5 inches and a sample standard deviation of 4 inches. Conduct a hypothesis test at the 0.05 level to see if the average trout length in this lake is less than mu=20 inches.

Answer 1

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u > 20
Alternative hypothesis: u < 20

Note that these hypotheses constitute a one-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method is a one-sample t-test.

Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).

SE = s / sqrt(n)

S.E = 0.5898
DF = n - 1

D.F = 45
t = (x - u) / SE

t = - 2.54

where s is the standard deviation of the sample, x is the sample mean, u is the hypothesized population mean, and n is the sample size.

The observed sample mean produced a t statistic test statistic of - 2.54.

Thus the P-value in this analysis is 0.007.

Interpret results. Since the P-value (0.007) is less than the significance level (0.05), hence we have to reject the null hypothesis.

From the above test we have sufficient evidence in the favor of the claim that the average trout length in this lake is less than mu=20 inches.