Question

A lumber company is making boards that are 2785 2785 millimeters tall. If the boards are too long they must be trimmed, and if they are too short they cannot be used. A sample of 39 39 boards is taken, and it is found that they have a mean of 2780.6 2780.6 millimeters. Assume a population standard deviation of 10 10 . Is there evidence at the 0.1 0.1 level that the boards are too short and unusable?

Answer 1

Given data

Sample mean x̅=2780.6

Population standard deviation σ=10 ,

Sample size n=39.

(1) Null and Alternative Hypothesis

The following null and alternative hypothesis need to be tested:

Ho:  µ = 2785

Ha:  µ < 2785

This corresponds to a left-tailed test, for which a z-test for one mean, with known population standard deviation will be used.

(2) Rejection Region

Based on the information provided, the significance level is a=0.10, and the critical value for a left-tailed test is zc= - 1.28 using Excel formula, =NORM.S.INV(0.1)

The rejection region for this left-tailed test is R = z<−1.28

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that z = -2.748 <zc ​=−1.28, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0.003 using Excel formula

=NORM.S.DIST(-2.748,TRUE)

since p = 0.003 <0.10, it is concluded that the null hypothesis is rejected.

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to support the claim that the population mean μ is less than 2785, at the 0.10 significance level. So, from above test we can say that boards are too short and unusable.