Question

Heavy bread? The mean weight of loaves of bread produced at the bakery where you work is supposed to be $1$pound. You are the supervisor of quality control at the bakery, and you are concerned that new employees are producing loaves that are too light. Suppose you weigh an SRS of bread loaves and find that the mean weight is $0.975$pound.

a. State appropriate hypotheses for performing a significance test. Be sure to define the parameter of interest.

b. Explain why there is some evidence for the alternative hypothesis.

c. The P-value for the test in part (a) is $0.0806$. Interpret the P-value.

d. What conclusion would you make at the $\alpha =0.01$ significance level?

Short answer

a. $H_0:\mu =1$and $H_1:\mu <1$

b. Sample mean of $0.975<1$pound and mean is less than one pound.

c. $8.06\%$possibility that mean weight of random sample of broad leaves $0.975$pound or more extreme, when mean weight is actually one pound.

d. There is no convincing proof that mean weight is less than one pound.

Step-by-step solution

Part (b) Step 1: Given Information

It is given that $\overline{x}=0.975$

Claim is mean is less than $1$.

Part (b) Step 2: Explanation

- The claim can be the null hypothesis or the alternative hypothesis.

- The null hypothesis statement is that population mean is equal to the value given in the claim.

- If the null hypothesis is the claim, then the alternative hypothesis statement is opposite of the null hypothesis.

$H_0:\mu =1$

$H_1:\mu <1$

Some proof for alternative hypothesis is there. It is because sample mean is less than one pound. It also corresponds with claim of alternative hypothesis that mean is less than one pound.

Part (a) Step 1: Given Information

Claim is mean is less than $1$pound.

Part (a) Step 2: Explanation

Null Hypothesis: Population value is equal to value in claim.

$H_0:\mu =1$

The claim can be the null hypothesis or the alternative hypothesis. The null hypothesis is that the population mean is equal to the value given in the claim.

If the null hypothesis is the claim, then the alternative hypothesis statement will be the opposite of the null hypothesis.

$H_1:\mu <1$ ($\mu$ is mean weight of all bread loaves.)

Part (c) Step 1: Given Information

$\overline{x}=0.975$

CLAIM: Mean is less than $1$.

$P=0.0806=8.06\%$

Part (c) Step 2: Explanation

As explained above

$H_0:\mu =1$

$H_1:\mu <1$

When null hypothesis is true, $P$value is probability of getting the sample results or extreme results.

Therefore, $8.06\%$ possibility that the mean weight of a simple random sample of bread loaves which is $0.975$ pound or more extreme, when the mean weight of all bread loaves is actually one pound.

Part (d) Step 1: Given Information

It is given that $\overline{x}=0.975$

$P=0.0806=8.06\%$

$\alpha =0.01$

CLAIM: Mean is less than one.

Part (d) Step 2: Explanation

From above:

$H_0:\mu =1$

$H_1:\mu <1$

Here, $0.0806>0.01\Rightarrow \text{Fail to eject}{H}_0$

So, there is no convincing proof that mean weight is less than one pound.