The director of manufacturing at a clothing factory needs to determine whether a new machine is producing a particular type of cloth according to the manufacturer's specifications, which indicate that the cloth should have a mean breaking strength of 70 pounds and a standard deviation of 3.5 pounds. A sample of 49 pieces of cloth reveals a sample mean breaking strength of 69.1 pounds.

a. Is there evidence that the machine is not meeting the manufacturer's specifications for mean breaking strength (use a 0.05 level of significance)?

b. Compute the p-value and interpret its meaning.

c. What is your answer in (a) if the standard deviation is 1.75 pounds?

d. What is your answer in (a) if the sample mean is 69 pounds and the standard deviation is 3.5 pounds?

**Solution: a.-) **We have a sample with _{}, and the sample mean is _{}. We are testing the hypothesis

_{}

We assume that the standard deviation is _{}. We build the test statistic _{} as

_{}

At the 0.05 level of significance, the critical value is _{}, and the rejection zone is

_{}

We notice that the statistic _{}doesn’t belong to the rejection zone, and therefore *we don’t reject the null hypothesis*. That means that there is no evidence that the machine is not meeting the manufacturer's specifications for mean breaking strength.

**b.-) **We have a two-tailed test, therefore we compute the P-value as_{}. If the P-value is less than the level of significance, we reject the null hypothesis; otherwise, we don’t have enough evidence to reject _{}.

**c.-) **With a new standard deviation _{} we recalculate the test statistic as

_{}

In this case, the test statistics does fall on the rejection zone

_{}

and therefore we reject the null hypothesis, meaning that there is evidence that the machine is not meeting the manufacturer's specifications for mean breaking strength (at the 0.05 level of significance).

**d.-) **We recalculate again _{}with _{}and _{}to obtain

_{}

and again there is evidence that the machine is not meeting the manufacturer's specifications for mean breaking strength, because _{}falls over the rejection zone.

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