Questions: A coin-operated drink machine was designed to discharge a mean of 6 fluid ounces of coffee per cup. In a test of the machine, the discharge amounts in 9 randomly chosen cups of coffee from the machine were recorded. The sample mean and sample standard deviation were 5.77 fluid ounces and 0.27 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, μ, differs from 6 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. (If necessary, consult a list of formulas.) (a) State the null hypothesis H0 and the alternative hypothesis H1. H0: μ=6 H1: μ ≠ 6 (b) Determine the type of test statistic to use. Degrees of freedom: 8 (c) Find the value of the test statistic. (Round to three or more decimal places.) 2.556 (d) Find the two critical values. (Round to three or more decimal places.) (e) Can we conclude that the mean discharge differs from 6 fluid ounces? Yes No

A coin-operated drink machine was designed to discharge a mean of 6 fluid ounces of coffee per cup. In a test of the machine, the discharge amounts in 9 randomly chosen cups of coffee from the machine were recorded. The sample mean and sample standard deviation were 5.77 fluid ounces and 0.27 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, μ, differs from 6 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. (If necessary, consult a list of formulas.) (a) State the null hypothesis H0 and the alternative hypothesis H1. H0: μ=6 H1: μ ≠ 6 (b) Determine the type of test statistic to use. Degrees of freedom: 8 (c) Find the value of the test statistic. (Round to three or more decimal places.) 2.556 (d) Find the two critical values. (Round to three or more decimal places.) (e) Can we conclude that the mean discharge differs from 6 fluid ounces? Yes No
Transcript text: A coin-operated drink machine was designed to discharge a mean of 6 fluid ounces of coffee per cup. In a test of the machine, the discharge amounts in 9 randomly chosen cups of coffee from the machine were recorded. The sample mean and sample standard deviation were 5.77 fluid ounces and 0.27 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, $\mu$, differs from 6 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. (If necessary, consult a list of formulas.) (a) State the null hypothesis $H_{0}$ and the alternative hypothesis $H_{1}$. \[ \begin{array}{l} H_{0}: \mu=6 \\ H_{1}: \mu \neq 6 \end{array} \] (b) Determine the type of test statistic to use. $\square$ \[ \text { Degrees of freedom: } 8 \] (c) Find the value of the test statistic. (Round to three or more decimal places.) \[ 2.556 \] (d) Find the two critical values. (Round to three or more decimal places.) $\square$ and $\square$ (e) Can we conclude that the mean discharge differs from 6 fluid ounces? Yes No
failed

Solution

failed
failed

Solution Steps

Step 1: State the Hypotheses

The null hypothesis \( H_0 \) and the alternative hypothesis \( H_1 \) are defined as follows: \[ H_0: \mu = 6 \] \[ H_1: \mu \neq 6 \]

Step 2: Determine the Test Statistic Type

Since the population standard deviation is unknown and the sample size is small (\( n < 30 \)), we will use a t-test. The degrees of freedom for this test is calculated as: \[ \text{Degrees of Freedom} = n - 1 = 9 - 1 = 8 \]

Step 3: Calculate the Test Statistic

First, we compute the standard error \( SE \): \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{0.27}{\sqrt{9}} = 0.09 \] Next, we calculate the t-test statistic: \[ t_{\text{test}} = \frac{\bar{x} - \mu_0}{SE} = \frac{5.77 - 6}{0.09} = -2.556 \]

Step 4: Determine the Critical Values

For a two-tailed test with a significance level of \( \alpha = 0.10 \) and \( 8 \) degrees of freedom, the critical t-values are: \[ t_{\text{critical, lower}} = -1.860 \] \[ t_{\text{critical, upper}} = 1.860 \]

Step 5: Make a Decision

The calculated test statistic \( t_{\text{test}} = -2.556 \) falls outside the range defined by the critical values \( (-1.860, 1.860) \). Therefore, we reject the null hypothesis.

Step 6: Conclusion

The p-value associated with the test statistic is \( P = 2 \times (1 - T(|z|)) = 0.034 \), which is less than the significance level \( \alpha = 0.10 \). Thus, we conclude that there is enough evidence to suggest that the mean discharge differs from \( 6 \) fluid ounces.

Final Answer

\[ \boxed{\text{Yes}} \]

Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
Identify a recognized category of defenses to liability for a negotiable instrument. - Symbolic defenses - Constructive defenses - Personal defenses - Equitable defenses
failedAnswered
A chemist dissolves 60 mg of pure sodium hydroxide in enough water to make up 70 mL of solution. Calculate the pH of the solution. (The temperature of the solution is 25 °C.) Be sure your answer has the correct number of significant digits.
failedAnswered
For the cost function C(x)=25600+400x+x^2. Use the formula C̄(x)=C(x)/x to compute the average cost. Then, find the zero of its derivative to find the production level x minimizing the average cost. The production level that minimizes the average cost is
failedAnswered
I'm sorry, but you didn't provide the content of the question from the given problems (3.39 and 3.40). Could you please provide the specific text of the question(s) you need extracted?
failedAnswered
shown in the