/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 133 Calculate the \(p\) -value for e... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 133

Calculate the \(p\) -value for each of the following hypothesis tests. a. \(H_{a}: \sigma^{2} \neq 20, n=15, \chi^{2} \star=27.8\) b. \(H_{a}: \sigma^{2}>30, n=18, \chi^{2} \star=33.4\) c. \(H_{a}: \sigma^{2} \neq 42,\) df \(=25, \chi^{2} \star=37.9\) d. \(H_{a}: \sigma^{2}<12,\) df \(=40, \chi^{2} \star=26.3\)

Short Answer

Expert verified
The p-values computation involves using a Chi-square distribution and applying the appropriate tail-test formula. Use a statistical table or computation software for the calculations. The final p-values will require further consideration based on a chosen significance level (\( \alpha\)), which isn't given in this case. The p-value is used to decide whether to reject or not reject the null hypothesis, with typically p鈮�0.05 being the evidence to reject \(H_{0}\).

Step by step solution

01

Define the Null Hypothesis (\(H_{0}\))

Firstly, identify the null hypothesis. The null hypothesis (\(H_{0}\)) is a statement about a population that will be tested. It often represents 'status quo' or 'no difference'. In these tests, they will compare to \(H_{a}\) as \(H_{0}:\sigma^{2}=20\) for a, \(H_{0}: \sigma^{2} \leq30\) for b, \(H_{0}: \sigma^{2}=42\) for c and \(H_{0}:\sigma^{2} \geq12\) for d.
02

Use Chi-Square Distribution

The formula to calculate the p-value depends on the type of hypothesis test. For all, we use the chi-square distribution because it's a test about the variance or standard deviation. You should remember that chi-square distribution is always positive, and it is skewed to the right.
03

Compute p-values

Next, calculate the p-values for each case. For a, we compute two tail-test: \(P( \chi^{2}_{14} > 27.8) + P( \chi^{2}_{14} < 14*20/27.8) \). For b, it's right tailed: \(P(\chi^{2}_{17} > 33.4)\). For c, again compute two tail-test: \(P(\chi^{2}_{25} > 37.9) + P(\chi^{2}_{25} < 25*42/37.9) \). And for d, it is left tailed: \(P(\chi^{2}_{40} < 26.3)\). These computations involve areas under the curve in chi-square distribution, for which statistical table or computation software can be used.
04

Interpret p-values

Finally, interpret the p-value for each case. A low p-value (typically 鈮� 0.05) indicates strong evidence against the null hypothesis, so you reject the null hypothesis. A large p-value (> 0.05) indicates weak evidence against the null hypothesis, so you fail to reject the null hypothesis.

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 91影视!

Key Concepts

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

chi-square distribution
The chi-square distribution is a fundamental concept in statistical testing, especially when dealing with variance hypotheses. This distribution is asymmetric, meaning it is skewed positively or to the right. It never takes negative values because it is defined only for non-negative numbers. The shape of the chi-square distribution depends on the degrees of freedom, denoted as "df".
  • As the degrees of freedom increase, the distribution becomes less skewed and more like a normal distribution.
  • It is primarily used in the context of hypothesis testing where the variance is under scrutiny.
This distribution is integral in tests like the chi-square test of independence or goodness-of-fit, where it helps decide the likelihood of a certain variable鈥檚 effect being observed if a particular null hypothesis is true.
null hypothesis
In hypothesis testing, the null hypothesis (\(H_{0}\) ) is a critical part of understanding and structuring experiments. Essentially, it is a statement that there is no effect or no difference, and it acts as a basis for statistical testing.
  • The null hypothesis is what statisticians try to reject or disprove with their data, hoping to demonstrate an alternate hypothesis.
  • For example, if you're testing a medication, the null hypothesis might state that the medication has no effect.
  • The outcome of hypothesis tests leads to decisions; either to reject the null hypothesis or to fail to reject it.
This rejection process relies heavily on p-values, which indicate how extreme the data are if the null hypothesis is true. Rejection of the null is usually sought after in attempts to provide evidence supporting an alternative hypothesis.
variance hypothesis testing
Variance hypothesis testing specifically deals with assessing variances within data groups or samples. It is crucial for determining the consistency or variability in datasets.
  • Standard methods involve using chi-square tests that analyze differences in sample variance as compared to a known variance.
  • In these tests, the null hypothesis often suggests that the variance equals a specific value, while the alternative hypothesis suggests variance differs.
  • For example, you may be testing if a new manufacturing process is producing the same level of product consistency as the old one, using variance tests.
The chi-square distribution comes into play as it helps determine whether observed data significantly deviate from what was expected under the null hypothesis. Preferably, software tools or statistical tables aid in computing these tests.
statistical significance
Statistical significance is a key outcome in hypothesis testing, often characterized by the p-value. It indicates whether a result is likely not due to just chance.
  • Commonly, a p-value below 0.05 is regarded as statistically significant, suggesting strong evidence against the null hypothesis.
  • Statistically significant results imply the data supports the alternative hypothesis enough to warrant consideration, potentially requiring actions or changes.
  • It's important because it sets thresholds for decision-making processes in both scientific inquiry and business applications.
However, one must be cautious; statistical significance does not imply practical significance. Even if results are statistically significant, they might not have substantial impact in real-world contexts.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

You hurry to the local emergency department in hopes of immediate, urgent care, only to find yourself waiting for what seems like hours. The manager of the large emergency department believes that his new procedures have substantially reduced the wait time for the average urgent care patient. He initiates a study to evaluate the wait time. The records of 18 randomly selected patients seen since the new procedures were put in place are checked, and the time between entering the emergency department and being seen by urgent care personnel was observed. The mean wait time was 17.82 minutes with a standard deviation of 5.68 minutes. Estimate the mean wait time using a \(99 \%\) confidence interval. Assume that wait times are normally distributed.

The National Highway Traffic Safety Administration found that, among the crashes with recorded times, EMS notification times exceeded 10 minutes in \(19.4 \%\) of rural fatal crashes. A random sample of 500 reported fatal crashes in Kentucky showed \(21.8 \%\) of the notification times exceeded 10 minutes. Construct the \(95 \%\) confidence interval for the true proportion of fatal crashes in Kentucky whose elapsed notification time exceeded 10 minutes.

A production process is considered out of control if the produced parts have a mean length different from \(27.5 \mathrm{mm}\) or a standard deviation that is greater than \(0.5 \mathrm{mm} .\) A sample of 30 parts yields a sample mean of \(27.63 \mathrm{mm}\) and a sample standard deviation of \(0.87 \mathrm{mm} .\) If we assume part length is a normally distributed variable, does this sample indicate that the process should be adjusted to correct the standard deviation of the product? Use \(\alpha=0.05\).

"You say tomato, burger lovers say ketchup!" According to a recent T.G.I. Friday's restaurants' random survey of 1027 Americans, approximately half \((47 \%)\) said that ketchup is their preferred burger condiment. The survey quoted a margin of error of plus or minus \(3.1 \% .\) a. Describe how this survey of 1027 Americans fits the properties of a binomial experiment. Specifically identify \(n,\) a trial, success, \(p,\) and \(x\). b. What is the point estimate for the proportion of all Americans who prefer ketchup on their burger? Is it a parameter or a statistic? c. Calculate the \(95 \%\) confidence maximum error of estimate for a binomial experiment of 1027 trials that results in an observed proportion of 0.47 d. How is the maximum error, found in part c, related to the \(3.1 \%\) margin of error quoted in the survey report? e. Find the \(95 \%\) confidence interval for the true proportion \(p\) based on a binomial experiment of 1027 trials that results in an observed proportion of 0.47.

Use a computer or calculator to find the \(p\) -value for the following hypothesis test: \(H_{o}: \sigma^{2}=7\) versus \(H_{a}: \sigma^{2} \neq 7\) if \(\chi^{2} \star=6.87\) for a sample of \(n=15\).
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.