A coin-operated drink machine was designed to discharge a mean of 7 fluid ounces of coffee per cup. In a test of the machine, the discharge amounts in 20 randomly chosen cups of coffee from the machine were recorded. The sample mean and sample standard deviation were 7.1 fluid ounces and 0.2 fluid ounces, respectively.
If we assume that the discharge amounts are approximately normally distributed, is there enough evidence, to conclude that the population mean discharge, , differs from 7 fluid ounces? Use the 0.10 level of significance.

Perform a two-tailed test. Then complete the parts below.

Carry your intermediate computations to three or more decimal places. (a)State the null hypothesis H0 and the alternative hypothesis H1 (b)Determine the type of test statistic to use. (c)Find the value of the test statistic. (Round to three or more decimal places.) (d)Find the two critical values. (Round to three or more decimal places.) (e)Can we conclude that the mean discharge differs from 7 fluid ounces? YES/No

To address this problem, we need to perform a hypothesis test using the given data. Here's how we can proceed step by step:

### Given:
- Population mean under the null hypothesis (\(\mu_0\)): 7 fluid ounces
- Sample mean (\(\bar{x}\)): 7.1 fluid ounces
- Sample standard deviation (\(s\)): 0.2 fluid ounces
- Sample size (\(n\)): 20
- Significance level (\(\alpha\)): 0.10

### (a) State the null hypothesis \(H_0\) and the alternative hypothesis \(H_1\):

- **Null Hypothesis** (\(H_0\)): \(\mu = 7\) fluid ounces (The population mean discharge is 7 fluid ounces.)
- **Alternative Hypothesis** (\(H_1\)): \(\mu \neq 7\) fluid ounces (The population mean discharge differs from 7 fluid ounces.)

### (b) Determine the type of test statistic to use:

Since the population standard deviation is not known and the sample size is small (\(n = 20\)), we will use the **t-test** statistic.

### (c) Find the value of the test statistic:

The test statistic for a t-test is given by the formula:
\[
t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}
\]

Plugging in the values:
\[
t = \frac{7.1 - 7}{\frac{0.2}{\sqrt{20}}} = \frac{0.1}{\frac{0.2}{4.472}} = \frac{0.1}{0.0447} \approx 2.236
\]

So, the value of the test statistic is approximately **2.236**.

### (d) Find the two critical values:

Since it's a two-tailed test with \(\alpha = 0.10\), the critical values will correspond to the 5% significance level in each tail. For 19 degrees of freedom (\(n - 1 = 20 - 1\)), we look up the t-distribution table (or use statistical software) to find the critical t-value.

For a two-tailed test with \( \alpha = 0.10 \), the critical t-values are:
\[
t_{crit} = \pm t_{0.05, 19}
\]

From the t-distribution table, \( t_{0.05, 19} \) is approximately **±1.729**.

### (e) Can we conclude that the mean discharge differs from 7 fluid ounces? (YES/NO)

To make this conclusion, compare the test statistic to the critical values:
- If \( t \) falls outside the range \([-1.729, 1.729]\), we reject the null hypothesis.
- If \( t \) falls inside the range \([-1.729, 1.729]\), we fail to reject the null hypothesis.

Since our test statistic \( t = 2.236 \) is greater than the upper critical value of 1.729, we reject the null hypothesis.

**Conclusion:**
Yes, there is enough evidence at the 0.10 significance level to conclude that the mean discharge differs from 7 fluid ounces.

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Would you like further details or have any questions?

### Related Questions:

1. What would the conclusion be if the significance level was set to 0.05?
2. How would the test change if the sample size were increased to 30?
3. What if the sample standard deviation was 0.1 instead of 0.2 fluid ounces?
4. What are Type I and Type II errors in the context of this test?
5. How do you interpret the p-value in hypothesis testing?
6. What is the impact of increasing the significance level on the conclusion of a hypothesis test?
7. How would the test be conducted if the population standard deviation was known?
8. What assumptions are made when performing a t-test?

**Tip:** Always ensure to check whether the assumptions of normality and equal variances are met before performing a t-test.

Explore a hypothesis test to determine if the mean discharge from a coffee machine differs from 7 fluid ounces. Includes step-by-step calculations using sample mean, standard deviation, and significance level.

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