/*! 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 35 For each of the following exampl... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 35

For each of the following examples of tests of hypothesis about \(\mu\), show the rejection and nonrejection regions on the \(t\) distribution curve. a. A two-tailed test with \(\alpha=.02\) and \(n=20\) b. A left-tailed test with \(\alpha=.01\) and \(n=16\) c. A right-tailed test with \(\alpha=.05\) and \(n=18\)

Short Answer

Expert verified
To find the rejection and non-rejection regions on the t-distribution curve, one must first determine the nature of the test - whether it's two-tailed, left-tailed or right-tailed. The value of \(\alpha\) helps in determining the area under the curve, and the degree of freedom, which is calculated by subtracting 1 from \(n\), aids in finding the critical t-value for respective tests. Once these values are known, the t-distribution curve can be plotted accordingly for each test type.

Step by step solution

01

Solving part a

In a two-tailed test with \(\alpha=.02\) and \(n=20\), there are two regions of rejection due to two tails on both extremes of the t-distribution curve. Here, \(\alpha\) is split between the two tails. Therefore, each tail of the curve will have an area of \(0.01\) underneath it. Use degrees of freedom, which is \(n-1 = 19\) for using the t-table to find the critical t-values in both directions.
02

Solving part b

For a left-tailed test with \(\alpha=.01\) and \(n=16\), the rejection region is only on the left side, as it's a left-tailed test. The area under the left tail will be \(0.01\). The degrees of freedom here will be \(n-1=15\). Use the t-table to find the critical t-value in the left direction.
03

Solving part c

In the case of a right-tailed test with \(\alpha=.05\) and \(n=18\), the rejection region is only on the right side. The area under the curve on the right tail will be \(0.05\). Similar to the previous steps, use degrees of freedom \(n-1=17\) to find the critical t-value in the right direction.

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.

t-distribution
When we perform a hypothesis test about a population mean, especially when the sample size is small or the population standard deviation is unknown, we use the t-distribution. The t-distribution is a critical tool because it helps us create more accurate conclusions about our population. It resembles the normal distribution but has heavier tails, allowing for more variability.
  • The shape of the t-distribution depends on the sample size, or more specifically, the degrees of freedom.
  • It becomes closer to a normal distribution as the sample size increases.
  • This distribution is used when conducting t-tests, a common type of hypothesis test.
Understanding t-distribution is crucial, as it lays the foundation for determining critical values and rejection regions in hypothesis testing.
critical value
In hypothesis testing, the critical value is a threshold that separates the rejection region from the non-rejection region on a t-distribution curve. It is a point that we compare our test statistic to. If the test statistic falls beyond the critical value, we have enough evidence to reject the null hypothesis.
  • Critical values depend on the significance level, or alpha (\(\alpha\)), and the degrees of freedom.
  • In a two-tailed test, there are two critical values because the rejection region is in two tails of the distribution.
  • In a one-tailed test, there is only one critical value, either to the left or the right.
Finding the correct critical value is a step that requires consulting the t-distribution table, which provides specific values given an alpha level and degrees of freedom.
rejection region
The rejection region is the area under the t-distribution curve where we reject the null hypothesis. This area is defined by the critical value or values, depending on whether it's a one-tailed or two-tailed test.
  • For a two-tailed test, the rejection region is split into two tails of the curve鈥攐ne on each end.
  • For a one-tailed test, the rejection region is present only on one side, either left or right.
The size of the rejection region depends on the significance level, \(\alpha\). A smaller alpha results in a smaller rejection region, making it harder to reject the null hypothesis. Visualizing this region on the t-distribution curve helps understand where the decision boundary lies.
degrees of freedom
Degrees of freedom (df) are an important concept in statistics, particularly in the context of t-distribution tests. They are essentially the number of values in a calculation that are free to vary. In our context, it's calculated as the sample size minus one, or \(df = n - 1\).
  • Degrees of freedom influence the shape of the t-distribution. As df increases, the t-distribution starts resembling a normal distribution more closely.
  • They determine which row of the t-table we should use when finding critical values.
Understanding degrees of freedom is vital because they directly affect the critical values and influence the decision in hypothesis testing.
alpha level
The alpha level, or significance level, is a crucial component in hypothesis testing. It represents the probability of making a Type I error, which is rejecting a true null hypothesis. Commonly used alpha levels include 0.05, 0.01, and 0.02.
  • An alpha level of 0.05, for example, indicates a 5% risk of concluding that a difference exists when there is none.
  • The alpha level determines the size of the rejection region. A smaller alpha leads to a smaller rejection region, making it tougher to reject the null hypothesis.
Choosing the right alpha level is an important decision, as it balances the risk of a Type I error with the need to detect a true effect. Adjusting alpha affects both critical values and the shape of the rejection region on the t-distribution curve.

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

Consider \(H_{0}: \mu=72\) versus \(H_{1}: \mu>72 . \Lambda\) random sample of 16 observations taken from this population produced a sample mean of \(75.2\). The population is normally distributed with \(\sigma=6\). a. Calculate the \(p\) -value. b. Considering the \(p\) -value of part a, would you reject the null hypothesis if the test were made at a significance level of \(.01\) ? c. Considering the \(p\) -value of part a, would you reject the null hypothesis if the test were made at a significance level of \(.025 ?\)

According to a survey by the College Board, undergraduate students at private nonprofit four-year colleges spent an average of \(\$ 1244\) on books and supplies in \(2014-2015\) (www.collegeboard.org). A recent random sample of 200 undergraduate college students from a large private nonprofit four-year college showed that they spent an average of \(\$ 1204\) on books and supplies during the last academic year. Assume that the standard deviation of annual expenditures on books and supplies by all such students at this college is \(\$ 200\). a. Find the \(p\) -value for the test of hypothesis with the alternative hypothesis that the annual mean expenditure by all such students at this college is less than \(\$ 1244\). Based on this \(p\) -value, would you reject the null hypothesis if the significance level is \(.025\) ? b. Test the hypothesis of part a using the critical-value approach and \(\alpha=.025 .\) Would you reject the null hypothesis? Explain.

Consider \(H_{0}: p=.45\) versus \(H_{1}: p<.45\). a. A random sample of 400 observations produced a sample proportion equal to .42. Using \(\alpha=.025\), would you reject the null hypothesis? b. Another random sample of 400 observations taken from the same population produced a sample proportion of .39. Using \(a=.025\), would you reject the null hypothesis? Comment on the results of parts a and \(\mathrm{b}\).

According to a 2014 CIRP Your First College Year Survey, \(88 \%\) of the first- year college students said that their college experience exposed them to diverse opinions, cultures, and values (www. heri.ucla.edu). Suppose in a recent poll of 1800 first-year college students, \(91 \%\) said that their college experience exposed them to diverse opinions, cultures, and values. Perform a hypothesis test to determine if it is reasonable to conclude that the current percentage of all firstyear college students who will say that their college experience exposed them to diverse opinions, cultures, and values is higher than \(88 \% .\) Use a \(2 \%\) significance level, and use both the \(p\) -value and the critical-value approaches.

A company claims that the mean net weight of the contents of its All Taste cereal boxes is at least 18 ounces. Suppose you want to test whether or not the claim of the company is true. Explain briefly how you would conduct this test using a large sample. Assume that \(\sigma=\) 25 ounce.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

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.