Problem 30 - Basic Statistics for Business and Economics with Student

Question

Dole Pineapple, Inc., is concerned that the 16 -ounce can of sliced pineapple is being overfilled. Assume the standard deviation of the process is .03 ounces. The quality-control department took a random sample of 50 cans and found that the arithmetic mean weight was 16.05 ounces. At the 5 percent level of significance, can we conclude that the mean weight is greater than 16 ounces? Determine the \(p\) -value.

Step-by-Step Solution

Answer

The mean weight is greater than 16 ounces with a p-value effectively 0.

Step 1: Define the Null and Alternative Hypothesis

The first step in hypothesis testing is to define the null hypothesis \((H_0)\) and the alternative hypothesis \((H_1)\). In this case:- Null Hypothesis \(H_0\): The mean weight of the cans is 16 ounces, \(\mu = 16\).- Alternative Hypothesis \(H_1\): The mean weight of the cans is greater than 16 ounces, \(\mu > 16\).

Step 2: Determine the Test Statistic

We use the formula for the test statistic for the sample mean when the population standard deviation is known:\[ z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \]Where:- \(\bar{x} = 16.05\) is the sample mean,- \(\mu = 16\) is the population mean according to the null hypothesis,- \(\sigma = 0.03\) is the population standard deviation,- \(n = 50\) is the sample size.Substituting the known values:\[ z = \frac{16.05 - 16}{\frac{0.03}{\sqrt{50}}} = \frac{0.05}{0.00424} \approx 11.79 \]

Step 3: Find the Critical Value and Compare

At a 5 percent level of significance with a one-tailed test, the critical value for \(z\) is approximately 1.645. Since our calculated \(z\)-value of 11.79 is much greater than 1.645, we reject the null hypothesis.

Step 4: Calculate the p-value

The \(p\)-value can be found using a standard normal \(z\)-table, which gives the probability that a standard normal random variable is less than a given value. For a \(z\)-value of 11.79, the \(p\)-value is extremely small, effectively 0, indicating strong evidence against the null hypothesis.

Step 5: Draw the Conclusion

Since the \(p\)-value is much less than the level of significance (0.05), we reject the null hypothesis and accept the alternative hypothesis that the mean weight is greater than 16 ounces.

Key Concepts

Null Hypothesis

In hypothesis testing, the null hypothesis is a statement that assumes there is no effect or no difference in the population from what is stated in the hypothesis. It is denoted by \( H_0 \). The purpose of the null hypothesis is to provide a basis for testing the actual outcome of the experiment. For example, in the Dole Pineapple exercise, the null hypothesis \( H_0 \) was that the mean weight of the cans is exactly 16 ounces, expressed as \( \mu = 16 \).
The null hypothesis is always tested with the intent of nullifying or disproving it. It is the hypothesis that you want to test against the observed data. It forms the basis for any statistical test of significance, where typically, we seek evidence to "reject" the null hypothesis in favor of an alternative hypothesis. The decision to reject or not depends on the results of statistical calculations like the \( p \)-value.
Here are some key points to remember about null hypothesis:

• It reflects the status quo or no change.
• The test is built to find evidence against \( H_0 \).
• Statistical testing is designed so that the null hypothesis is "guilty until proven innocent."
• The null hypothesis is accepted as true until evidence shows otherwise.

Alternative Hypothesis

The alternative hypothesis represents a new claim, different from the null hypothesis, and symbolized by \( H_1 \) or \( H_a \). It is what the researcher aims to support and its acceptance is contingent upon the rejection of the null hypothesis. In our exercise, the alternative hypothesis \( H_1 \) posited that the mean weight of the cans is greater than 16 ounces, expressed as \( \mu > 16 \).
Unlike the null hypothesis, the alternative hypothesis is typically what the researcher believes to be true. It indicates the presence of an effect or difference implying that something significant is happening in the data being analyzed.
Key aspects of the alternative hypothesis include:

• It represents the outcome the researcher is testing for.
• It can involve a parameter being greater than, less than, or different from the parameter in \( H_0 \).
• The goal of the test is to provide enough evidence to reject \( H_0 \) in support of \( H_1 \).
• Decisions made using \( p \)-values and test statistics determine if \( H_1 \) is accepted.

p-value

The \( p \)-value is a crucial component in hypothesis testing that helps determine the strength of the evidence against the null hypothesis. It is the probability of observing a test statistic as extreme, or more extreme, given that the null hypothesis is true. In simple terms, it tells us how likely it is to get the observed data if the null hypothesis \( H_0 \) were true.
In our example, after computing the \( z \)-value, we found the \( p \)-value to be nearly zero. This indicates strong evidence against the null hypothesis and supports the conclusion that the mean weight is indeed greater than 16 ounces.
Understanding \( p \)-values:

• A small \( p \)-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis, so you reject \( H_0 \).
• A large \( p \)-value (> 0.05) indicates weak evidence against \( H_0 \), and you fail to reject the null hypothesis.
• It quantifies the results, helping researchers judge the significance of the results.
• The \( p \)-value complements the confidence level and can reinforce conclusions from statistical tests.