91Ó°ÊÓ

At Farmer's Dairy, a machine is set to fill 32 -ounce milk cartons. However, this machine does not put exactly 32 ounces of milk into each carton; the amount varies slightly from carton to carton but has a normal distribution. It is known that when the machine is working properly, the mean net weight of these cartons is 32 ounces. The standard deviation of the milk in all such cartons is always equal to \(.15\) ounce. The quality control inspector at this company takes a sample of 25 such cartons every week, calculates the mean net weight of these cartons, and tests the null hypothesis, \(\mu=32\) ounces, against the alternative hypothesis, \(\mu \neq 32\) ounces. If the null hypothesis is rejected, the machine is stopped and adjusted. A recent sample of 25 such cartons produced a mean net weight of \(31.93\) ounces a. Calculate the \(p\) -value for this test of hypothesis. Based on this \(p\) -value, will the quality control inspector decide to stop the machine and readjust it if she chooses the maximum probability of a Type I error to be \(.01 ?\) What if the maximum probability of a Type I error is .05? b. Test the hypothesis of part a using the critical-value approach and \(\alpha=.01\). Does the machine need to be adjusted? What if \(\alpha=.05 ?\)

Step by step solution

01

State the hypotheses

The null hypothesis \(H_0: \mu = 32\) means the machine is working properly, while the alternative hypothesis \(H_1: \mu \neq 32\) implies it is not.

02

Calculate test statistic

The test statistic in this case is a z-score, as we have population standard deviation. Test statistic can be calculated using the formula:\( z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}} \), where \(\bar{x}\) is the sample mean, \(\mu_0\) is the population mean, \sigma is the standard deviation, and n is the sample size. Substituting the given values, we get \( z = \frac{31.93 - 32}{0.15/\sqrt{25}} \).

03

Find the p-value

Using a z-table or statistical software to find the two-tailed p-value associated with the calculated z-score. Remember that this is a two-tailed test as the alternative hypothesis is \(\mu \neq 32\).

04

Decision based on p-value

If the calculated p-value is less than the significance level (\(\alpha\)), the null hypothesis is rejected. We compare the p-value with \(\alpha=0.01\) and \(\alpha=0.05\) to conclude whether the machine needs adjustment in both cases.

05

Repeat the test using the critical value approach

For \(\alpha = 0.01\) and \(\alpha = 0.05\), determine the critical z-scores using a z-table or statistical software. For a two-tailed test, you have to consider both positive and negative critical z-values. Compare the calculated z-score with the critical z-scores to make a conclusion.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Null Hypothesis

In hypothesis testing, the **Null Hypothesis** is a foundational concept. It represents the default position or the statement that there is no effect or no difference. In the context of our exercise, the null hypothesis (\(H_0\)) is that the mean weight of the milk cartons filled by the machine is exactly 32 ounces. This implies that the machine is functioning as expected.
Adopting the null hypothesis is akin to assuming there are no changes unless there's significant evidence to prove otherwise. In essence, the null hypothesis stands as the position you seek to test against. It is important to craft this hypothesis carefully to ensure you're testing the right question.

• The null hypothesis is typically denoted as \(H_0\).
• It often assumes no effect or no difference.
• It's the hypothesis that one seeks to possibly reject with statistical evidence.

Alternative Hypothesis

The **Alternative Hypothesis** signifies a statement that opposes the null hypothesis. In the milk carton example, the alternative hypothesis (\(H_1\)) suggests that the mean weight of the cartons is not equal to 32 ounces, which implies something is off with the machine.
This hypothesis proposes potential changes or effects in the population mean that requires scrutiny. It's vital as it provides the direction and focus of the research, guiding the analysis toward detecting deviations from the norm.

• The alternative hypothesis is represented by \(H_1\)
• It indicates the presence of an effect or a difference.
• It is what you look to "prove" in statistical terms by rejecting the null hypothesis.

p-value

The **p-value** is a critical aspect of hypothesis testing. It represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true.
In simpler terms, the p-value helps you gauge the strength of the evidence against the null hypothesis. If it's low, it suggests that the observed data is unlikely under the null hypothesis, prompting a potential rejection. In our exercise, the p-value is compared with different significance levels (e.g., 0.01 and 0.05) to determine if the machine should be adjusted.

• A low p-value indicates strong evidence against the null hypothesis.
• It's a measure of the strength of the evidence presented by the data.
• The smaller the p-value, the stronger the evidence against \(H_0\)

z-score

A **z-score** is a statistical measurement that describes a value's relation to the mean of a group of values, measured in terms of standard deviations. In the context of hypothesis testing, it tells us how far our sample mean is from the hypothesized population mean in standard deviation units.
For the milk carton exercise, calculating the z-score helps determine how far the sample's mean weight (31.93 ounces) deviates from the expected mean (32 ounces). It's a step towards determining whether this deviation is significant or not. The formula used for calculation is \(z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}\).

• A high absolute z-score indicates a statistically significant result.
• Z-scores can be compared to critical values in a z-table to determine significance.
• It helps translate a sample's findings back to the concept of normal distribution.

Significance Level

The **Significance Level** (commonly denoted as \(\alpha\)) is the threshold used in hypothesis testing to determine whether to reject the null hypothesis. It represents the probability of committing a Type I error, which is mistakenly rejecting a true null hypothesis.
Choosing a significance level is essential, as it determines the extent of evidence required to reject the null hypothesis. In typical applications, values like 0.05 or 0.01 are frequently used. These signify a 5% or 1% risk of concluding that an effect exists when it actually doesn't. In the milk machine example, comparing the p-value to the significance level allows us to decide if the machine's settings require re-adjustment.

• Denoted by \(\alpha\)
• Common choices for \(\alpha\) are 0.05, 0.01, and 0.10.
• A lower \(\alpha\) provides more stringent criteria for rejecting \(H_0\)
• It controls the balance between sensitivity and specificity in hypothesis tests.