Question

a can of soda is labeled as containing 16 fluid ounces. the quality control manager wants to verify that the filling machine is neither over-filling nor under-filling the cans. complete parts (a) through (d) below.

(a) determine the null and alternative hypotheses that would be used to determine if the filling machine is calibrated correctly.

$h_0$: $\boldsymbol{
abla}$ $\boldsymbol{
abla}$ $square$
$h_1$: $\boldsymbol{
abla}$ $\boldsymbol{
abla}$ $square$
(type integers or decimals. do not round.)

(b) the quality control manager obtains a sample of 75 cans and measures the contents. the sample evidence leads the manager to reject the null hypothesis. write a conclusion for this hypothesis test.

there $\boldsymbol{
abla}$ sufficient evidence to conclude that the machine is out of calibration.

(c) suppose, in fact, the machine is not out of calibration. has a type i or type ii error been made?

a $\boldsymbol{
abla}$ has been made since the sample evidence led the quality-control manager to $\boldsymbol{
abla}$ the null hypothesis, when the $\boldsymbol{
abla}$ is true.

(d) management has informed the quality control department that it does not want to shut down the filling machine unless the evidence is overwhelming that the machine is out of calibration. what level of significance would you

Explanation:

Part (a)

Step1: Define Null Hypothesis

The null hypothesis \( H_0 \) is a statement of no effect or no difference. Here, we assume the machine fills cans with 16 fluid ounces (correct calibration). So \( H_0: \mu = 16 \).

Step2: Define Alternative Hypothesis

The alternative hypothesis \( H_1 \) is what we test against the null. Since we want to check if it's over - filling or under - filling (not equal to 16), \( H_1: \mu
eq 16 \).

When we reject the null hypothesis, we have enough evidence to support the alternative hypothesis. The alternative hypothesis here is that the machine is not filling with 16 fluid ounces (out of calibration). So the conclusion is that there is sufficient evidence.

Step1: Recall Error Types

• Type I error: Rejecting a true null hypothesis.
• Type II error: Failing to reject a false null hypothesis.

Step2: Analyze the Situation

The null hypothesis (\( H_0: \mu = 16 \)) is true (machine is not out of calibration), but we rejected the null hypothesis. So this is a Type I error.

Answer:

\( H_0: \boldsymbol{\mu = 16} \), \( H_1: \boldsymbol{\mu
eq 16} \)

Part (b)