An automatic coffee machine dispenses cups of coffee whose volume per cup varies normally with the mean μ = 10 oz. A quality-control researcher randomly selects 8 cups of coffee from the machine and finds that in this sample the mean volume is 9.92 oz. and the standard deviation is 0.23 oz.We want to determine if these data provide enough evidence to conclude that the mean volume per cup is below the target level.
Question
An automatic coffee machine dispenses cups of coffee whose volume per cup varies normally with the mean μ = 10 oz. A quality-control researcher randomly selects 8 cups of coffee from the machine and finds that in this sample the mean volume is 9.92 oz. and the standard deviation is 0.23 oz.We want to determine if these data provide enough evidence to conclude that the mean volume per cup is below the target level.
Solution
To determine if the data provides enough evidence to conclude that the mean volume per cup is below the target level, we can perform a one-sample t-test. Here are the steps:
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State the hypotheses: The null hypothesis is that the mean volume per cup is equal to the target level (μ = 10 oz). The alternative hypothesis is that the mean volume per cup is less than the target level (μ < 10 oz).
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Calculate the test statistic: The test statistic for a one-sample t-test is calculated as (sample mean - population mean) / (sample standard deviation / sqrt(sample size)). In this case, it would be (9.92 - 10) / (0.23 / sqrt(8)).
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Determine the critical value: The critical value for a one-tailed t-test with a significance level of 0.05 and 7 degrees of freedom (sample size - 1) can be found in a t-distribution table or calculated using a statistical software.
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Compare the test statistic to the critical value: If the test statistic is less than the critical value, we reject the null hypothesis in favor of the alternative hypothesis. This would provide enough evidence to conclude that the mean volume per cup is below the target level.
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Calculate the p-value: The p-value is the probability of obtaining a result as extreme as the observed data, assuming the null hypothesis is true. If the p-value is less than the significance level (typically 0.05), we reject the null hypothesis.
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Draw a conclusion: Based on the comparison of the test statistic to the critical value and the p-value, we can draw a conclusion about whether the data provides enough evidence to conclude that the mean volume per cup is below the target level.
Similar Questions
An automatic coffee machine dispenses cups of coffee whose volume per cup varies normally with the mean μ = 10 oz. A quality-control researcher randomly selects 8 cups of coffee from the machine and finds that in this sample the mean volume is 9.92 oz. and the standard deviation is 0.23 oz.Below is the output:Which of the following represents the correct conclusion we can make based on the output (and at the usual significance level of 0.05)? The data provide enough evidence to reject H0 and to conclude that the mean volume per cup is lower than the target level of 10 oz. The data provide enough evidence to accept H0 and to conclude that the mean volume per cup is at the target level of 10 oz. The data do not provide enough evidence to reject H0, so we accept it, and conclude that the mean volume per cup is at the target level of 10 oz. The data do not provide enough evidence to reject H0, nor to conclude that the mean volume per cup is lower than the target level of 10 oz.Question 4
n automatic coffee machine dispenses cups of coffee whose volume per cup varies normally with the mean μ = 10 oz. A quality-control researcher randomly selects 8 cups of coffee from the machine and finds that in this sample the mean volume is 9.92 oz. and the standard deviation is 0.23 oz.In this problem, we have made slight changes to the "coffee machine" story above. In which of the options below is the change such that it would be inappropriate to conduct a hypothesis test for: H0: μ = 10 vs. Ha: μ < 10. Check all that apply. An automatic coffee machine dispenses cups of coffee whose volume per cup varies according to a distribution with mean μ = 10 oz. A quality-control researcher randomly selects 8 cups of coffee from the machine and finds that in this sample the mean volume is 9.92 oz. and the standard deviation is 0.23 oz. An automatic coffee machine dispenses cups of coffee whose volume per cup varies according to a distribution with mean μ = 10 oz. A quality-control researcher randomly selects 40 cups of coffee from the machine and finds that in this sample the mean volume is 9.92 oz. and the standard deviation is 0.23 oz. An automatic coffee machine dispenses cups of coffee whose volume per cup varies normally with mean μ = 10 oz. and standard deviation σ = 0.23 oz. A quality-control researcher randomly selects 8 cups of coffee from the machine and finds that in this sample the mean volume is 9.92 oz. An automatic coffee machine dispenses cups of coffee whose volume per cup varies according to a distribution with mean μ = 10 oz. and standard deviation σ = 0.23 oz. A quality-control researcher randomly selects 8 cups of coffee from the machine and finds that in this sample the mean volume is 9.92 oz
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A manufacturer of chocolate candies uses machines to package candies as they move along a filling line.Although the packages are labeled as 8 ounces,the company wants the packages to contain a mean of 8.17 ounces so that virtually none of the packages contain less than 8 ounces.A sample of 50 packages is selected periodically,and the packaging process is stopped if there is deviation of 0.059 ounce.Complete parts (a)and(b) State the null and alternative hypotheses. H0μ= 8.17 H1:μ≠8.17 Determine the test statistic. The test statistic is
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