Question

The thickness of wooden boards, X (in inches), after being processed by a saw mill can be modeled by a N(µ, σ2 ) distribution. A random sample of 9 boards yielded x¯ = 1.02 inches and s = 0.02 inches. We wish to test H0 : µ = 1 versus Ha : µ 6= 1 at the α = 0.05 significance level. Which of the following is/are true? (a) µ significantly differs from 1 (b) The test statistic does not fall in the rejection region (c) The p−value for this test is in (0.005, 0.01) (d) All of the above (e) Both (a) and (c)

Answer 1

Let be the true average thickness of wooden boards.

The hypotheses are

We have the following information from the sample

n=9 is the sample size

inches is the sample average thickness

is the sample standard deviation of thickness

We estimate the population standard deviation using the sample as

The estimated standard error of means is

The hypothesized value of the mean thickness is

Since the sample size n=9 is less than 30 and we do not know the population standard deviation, assuming a normal distribution for the population, we can say that the sampling distribution of means has t distribution.

That is, we will do a 1 sample t tets for means

The test statistic is

The degrees of freedom are n-1=9-1=8

This is a 2 tailed test (The alternative hypothesis has "not equal to").

The right tail critical value for significance level is

Using the t tables for df=8, and the area under the right tail=0.025, we get

The critical values are -2.306, + 2.306.

The rejection region lies outside the interval ( -2.306 to + 2.306). That is we will reject the null hypothesis, if the test statistic lies outside the interval ( -2.306 to + 2.306).

Here, the test statistic is 3 and it lies outside the interval ( -2.306 to + 2.306). Hence we reject the null hypothesis.

We can conclude that the mean thickness significantly differs from 1

The test statistic falls in the rejection region. µ significantly differs from 1

Among the options, a) is correct, b is ruled out. Now we need ot check if c) is correct

We will get the p-value.

This is a 2 tailed test. The p-value is the sum of the area under both the tails

Using technology (calculator or Excel function =T.DIST.2T(3,8)), we get the p-value=0.017. the p−value for this test is not in (0.005, 0.01)

If you cannot use technology, then

Using the t tables for df=8 and the area under both the tails=0.01, we get a value of 3.355 and for df=8 and the area under both the tails=0.005, we get a value of 3.833. The test statistic does not lie between 3.355 and 3.833 and hence the p−value for this test is not in (0.005, 0.01)

So among the options only a) is correct

ans: (a) µ significantly differs from 1