/*! 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 53 A hypothesis will be used to tes... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 53

A hypothesis will be used to test that a population mean equals 10 against the alternative that the population mean is greater than 10 with unknown variance. What is the critical value for the test statistic \(T_{0}\) for the following significance levels? (a) \(\alpha=0.01\) and \(n=20\) (b) \(\alpha=0.05\) and \(n=12\) (c) \(\alpha=0.10\) and \(n=15\)

Short Answer

Expert verified
The critical values are 2.539, 1.796, and 1.345 for parts (a), (b), and (c) respectively.

Step by step solution

01

Identify the Type of Test

The test involves comparing a sample mean to a population mean with an unknown variance, indicating a one-sample t-test with an upper-tailed alternative hypothesis. We need to find the critical t-values for different sample sizes and significance levels.
02

Determine Degrees of Freedom

The degrees of freedom for a t-test is given by \( df = n - 1 \). For (a) \( n = 20 \), the degrees of freedom \( df = 19 \). For (b) \( n = 12 \), \( df = 11 \). For (c) \( n = 15 \), \( df = 14 \).
03

Consult the T-Distribution Table

Using the degrees of freedom, we look up the critical t-values from a t-distribution table for the given significance levels (upper-tailed test).
04

Find Critical Value for \(\alpha=0.01\) and \(df=19\)

For significance level \( \alpha = 0.01 \) and \( 19 \) degrees of freedom, the critical value \( t_{19, 0.01} \approx 2.539 \).
05

Find Critical Value for \(\alpha=0.05\) and \(df=11\)

For significance level \( \alpha = 0.05 \) and \( 11 \) degrees of freedom, the critical value \( t_{11, 0.05} \approx 1.796 \).
06

Find Critical Value for \(\alpha=0.10\) and \(df=14\)

For significance level \( \alpha = 0.10 \) and \( 14 \) degrees of freedom, the critical value \( t_{14, 0.10} \approx 1.345 \).

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.

One-Sample T-Test
A one-sample t-test is a statistical method used to determine if the mean of a single sample is significantly different from a known or hypothesized population mean. It is particularly useful when the population standard deviation is unknown, and the sample size is small鈥攖ypically fewer than 30 observations. The essence of this test is to compare the observed sample data to the hypothesis about the population.Here's how it works:
  • Hypotheses Setup: Usually, we set up two hypotheses: the null hypothesis (\( H_0 \)), which represents the status quo (i.e., the population mean is equal to a specified value), and the alternative hypothesis (\( H_a \)), which suggests a deviation from this norm.
  • T-Statistic Calculation: The t-statistic is calculated using the sample mean, hypothesized population mean, sample standard deviation, and sample size. It's given by the formula:\[T = \frac{\bar{X} - \mu}{S/\sqrt{n}}\]where \( \bar{X} \) is the sample mean, \( \mu \) is the population mean, \( S \) is the sample standard deviation, and \( n \) is the sample size.
  • Decision-Making: By comparing the calculated t-value against the critical value from the t-distribution (given the specified significance level and degrees of freedom), we decide whether to reject or fail to reject the null hypothesis.
Critical Value
The critical value is a threshold in hypothesis testing that determines the boundary beyond which we consider the test statistic to be significant. It helps us decide whether to reject the null hypothesis. The critical value is derived from the selected significance level and degrees of freedom, based on the t-distribution.Understanding critical values involves:
  • Significance Level Relation: Cirtical values depend on the chosen alpha (\( \alpha \)) level, which is the probability of committing a Type I error鈥攚rongly rejecting a true null hypothesis.
  • Direction of Test: Since one-sample t-tests can be one-tailed (testing only one direction) or two-tailed (testing both directions), the critical value region can differ. Our exercise focuses on a one-tailed test, seeking whether the sample mean is greater than the hypothesized mean.
  • Usage of t-Table: With the known degrees of freedom and chosen significance level, we look into t-tables to find the numerical critical value. This value is then used to gauge the meaningfulness of the computed t-statistic.
Significance Level
The significance level, denoted as alpha (\( \alpha \)), represents the threshold for determining statistical significance in hypothesis tests. It quantifies the risk we're willing to take for rejecting a true null hypothesis, known as a Type I error.In more detail:
  • Setting Alpha: Common significance levels are 0.01, 0.05, and 0.10. The value is chosen based on the context of the test and the stringency desired in decision-making. A smaller alpha means a stricter criterion for significance.
  • Impact on Critical Value: A lower significance level results in a higher critical value, making it harder to reject the null hypothesis because more evidence from the data is required.
  • Relation to P-Value: The significance level also interacts with the p-value, a more precise indicator of evidence against the null hypothesis. The null hypothesis is rejected only when the p-value is less than \( \alpha \).
Degrees of Freedom
Degrees of freedom, often abbreviated as df, are crucial in the context of the t-distribution. They represent the number of values in a calculation that are free to vary and are essential in finding critical values from statistical tables.Here's how they work:
  • Calculation Method: In our one-sample t-test scenario, degrees of freedom are calculated as \( df = n - 1 \), where \( n \) is the sample size. It helps adjust for model complexity.
  • Effect on Distribution: Degrees of freedom influence the shape of the t-distribution. As df increases, the t-distribution becomes more like a normal distribution. For smaller df, the tails of the t-distribution are heavier.
  • Practical Application: When consulting t-tables to find critical values for hypothesis tests, knowing the degrees of freedom allows you to find the appropriate critical t-value needed for making decisions about the null hypothesis.
We'll use these concepts to understand how degrees of freedom are applied in hypothesis testing and how they help shape our confidence in the final results.

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

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a sample from an exponential distribution with parameter \(\lambda\). It can be shown that \(2 \lambda \sum_{i=1}^{n} X_{i}\) has a chi-square distribution with \(2 n\) degrees of freedom. Use this fact to devise a test statistic and critical region for \(H_{0}: \lambda=\lambda_{0}\) versus the three usual alternatives.

A hypothesis will be used to test that a population mean equals 7 against the alternative that the population mean does not equal 7 with known variance \(\sigma .\) What are the critical values for the test statistic \(Z_{0}\) for the following significance levels? (a) 0.01 (b) 0.05 (c) 0.10

The life in hours of a battery is known to be approximately normally distributed with standard deviation \(\sigma=1.25\) hours. A random sample of 10 batteries has a mean life of \(\bar{x}=40.5\) hours. (a) Is there evidence to support the claim that battery life exceeds 40 hours? Use \(\alpha=0.05 .\) (b) What is the \(P\) -value for the test in part (a)? (c) What is the \(\beta\) -error for the test in part (a) if the true mean life is 42 hours? (d) What sample size would be required to ensure that \(\beta\) does not exceed 0.10 if the true mean life is 44 hours? (e) Explain how you could answer the question in part (a) by calculating an appropriate confidence bound on life.

A new type of tip can be used in a Rockwell hardness tester. Eight coupons from test ingots of a nickelbased alloy are selected, and each coupon is tested using the new tip. The Rockwell C-scale hardness readings are 63,65 , \(58,60,55,57,53,\) and \(59 .\) Do the results support the claim that the mean hardness exceeds 60 at a 0.05 level?

A random sample of 500 registered voters in Phoenix is asked if they favor the use of oxygenated fuels yearround to reduce air pollution. If more than 315 voters respond positively, we will conclude that at least \(60 \%\) of the voters favor the use of these fuels. (a) Find the probability of type I error if exactly \(60 \%\) of the voters favor the use of these fuels. (b) What is the type II error probability \(\beta\) if \(75 \%\) of the voters favor this action?
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Mechanics 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.