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The article "'Evaluating Variability in Filling Operations" (Food Tech., 1984: 51-55) describes two different filling operations used in a ground-beef packing plant. Both filling operations were set to fill packages with \(1400 \mathrm{~g}\) of ground beef. In a random sample of size 30 taken from each filling operation, the resulting means and standard deviations were \(1402.24 \mathrm{~g}\) and \(10.97 \mathrm{~g}\) for operation 1 and \(1419.63 \mathrm{~g}\) and \(9.96 \mathrm{~g}\) for operation 2 . a. Using a \(.05\) significance level, is there sufficient evidence to indicate that the true mean weight of the packages differs for the two operations? b. Does the data from operation 1 suggest that the true mean weight of packages produced by operation 1 is higher than \(1400 \mathrm{~g}\) ? Use a \(.05\) significance level.

Step by step solution

01

Define Hypotheses for Part (a)

We want to test if there is a difference in the true mean weights of the packages between the two operations. - Null Hypothesis (H_0): \( \mu_1 = \mu_2 \) (The mean weights are equal).- Alternative Hypothesis (H_a): \( \mu_1 eq \mu_2 \) (The mean weights are different).

02

Collect Given Data for Part (a)

We have the following statistics for the two operations:- Operation 1: \( \bar{x}_1 = 1402.24 \), \( s_1 = 10.97 \), \( n_1 = 30 \).- Operation 2: \( \bar{x}_2 = 1419.63 \), \( s_2 = 9.96 \), \( n_2 = 30 \).

03

Compute the Test Statistic for Part (a)

Use the two-sample t-test for equality of means:\[t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} = \frac{1402.24 - 1419.63}{\sqrt{\frac{10.97^2}{30} + \frac{9.96^2}{30}}}\]Calculate this value to determine the test statistic.

04

Compute Degrees of Freedom for Part (a)

The degrees of freedom for the t-test can be calculated using the formula:\[v = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1-1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2-1}}\]Perform this calculation to find the correct degrees of freedom.

05

Decision Rule for Part (a)

Determine the critical t-value from the t-distribution table at \( \alpha = 0.05\) with calculated degrees of freedom. Compare the computed test statistic from Step 3 with the critical value.

06

Conclusion for Part (a)

If the test statistic is beyond the critical t-value, reject the null hypothesis. Otherwise, do not reject it. - Conclusion: There is enough evidence to conclude that the true mean weights of packages from the two operations differ, or not.

07

Define Hypotheses for Part (b)

We will test if the mean weight for operation 1 is greater than 1400 grams.- Null Hypothesis (H_0): \( \mu_1 = 1400 \)- Alternative Hypothesis (H_a): \( \mu_1 > 1400 \)

08

Compute the Test Statistic for Part (b)

Use the one-sample t-test for operation 1:\[t = \frac{\bar{x}_1 - 1400}{\frac{s_1}{\sqrt{n_1}}} = \frac{1402.24 - 1400}{\frac{10.97}{\sqrt{30}}}\]Calculate this to get the test statistic.

09

Decision Rule for Part (b)

Find the critical t-value for a one-tailed test at the 0.05 significance level with 29 degrees of freedom. Compare this value with the calculated test statistic from Step 8.

10

Conclusion for Part (b)

If the test statistic exceeds the critical value, reject the null hypothesis. Otherwise, do not reject it. - Conclusion: There is enough evidence to suggest that the true mean weight for operation 1 is greater than 1400 grams, or not.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

t-test

A t-test is a statistical test used to evaluate hypotheses about means. It compares the sample means to determine whether there is a significant difference between them. The t-test is especially useful when dealing with small sample sizes, typically below 30, and assumes that the data follows a normal distribution.

There are different types of t-tests, but the two most common ones are:

• Independent two-sample t-test: It is used to compare the means of two independent groups, as in the case of different filling operations in our example. It tests if the difference between the sample means is significant.
• One-sample t-test: It compares the mean of a single sample to a known value or theoretical expectation. In our context, it checks if the mean of operation 1 exceeds 1400 grams.

The formula to calculate the t-statistic is \[t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}\]This result tells you "how far" your sample mean is from the null hypothesis mean—in standard error units. A larger absolute value of t suggests a significant difference.

significance level

The significance level, often denoted by the Greek letter \(\alpha\), is a threshold set by the researcher to determine the threshold for rejecting the null hypothesis. It represents the probability of making a Type I error, which means rejecting the null hypothesis when it is actually true. Commonly, a significance level of 0.05 is used, implying a 5% risk of concluding that a difference exists when there is none.

When you perform a t-test:

• The significance level helps you decide if your test statistic is extreme enough to reject the null hypothesis.
• If your calculated p-value is less than \(\alpha\), it indicates strong enough evidence against the null hypothesis, prompting you to reject it.
• A significance level of 0.05 was used in the given problem, meaning the risk of error should stay below 5%.

This risk level is intentionally kept low as a standard to ensure that decisions made based on the test results are reliable, accounting for minor random differences in sample versus population data.

degrees of freedom

Degrees of freedom are a vital concept in statistics that impact the shape of the t-distribution used in hypothesis testing. Essentially, degrees of freedom (often abbreviated as df) refer to the number of independent values or quantities that can vary in an analysis without breaking any mathematical constraints.

In the context of a t-test:

• For a two-sample t-test like in our example, degrees of freedom are calculated using the formula: \[v = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1-1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2-1}}\]which considers both samples' variances and sizes.
• For simple one-sample tests, degrees of freedom are \(n - 1\), where \(n\) is the sample size.

Understanding the degrees of freedom helps control the "spread" of the t-distribution; more degrees typically mean a more narrow and precise distribution. This precision impacts the critical values used for deciding whether to reject the null hypothesis.