Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Give as much information as you can about the \(P\) -value of a \(t\) test in each of the following situations: a. Two-tailed test, \(\mathrm{df}=9, t=0.73\) b. Upper-tailed test, \(\mathrm{df}=10, t=-0.5\) c. Lower-tailed test, \(n=20, t=-2.1\) d. Lower-tailed test, \(n=20, t=-5.1\) e. Two-tailed test, \(n=40, t=1.7\)

Short Answer

Expert verified
The P-values for the given situations are approximately: a. 1, b. 0.69, c. 0.025, d. 0.001, e. 0.096

Step by step solution

01

Determine P-value for Two-tailed test, df=9, t=0.73

To find the p-value, we must look at a t-distribution table for the given degrees of freedom (df = 9). In this case, we find that t=0.73 corresponds to a probability in the t-distribution table of about 0.77, which is the one-tailed p-value. Because the test is two-tailed, we must double this to get the two-tailed P-value: 0.77*2 = 1.54. Since a probability cannot exceed 1, we interpret this as a P-value of 1, indicating weak evidence against the null hypothesis.
02

Determine P-value for Upper-tailed test, df=10, t=-0.5

Looking at the t-distribution table for df = 10, we find that t = -0.5 corresponds to a probability in the table of about 0.69, which is the two-tailed p-value. Because the test is upper-tailed (or one-tailed), the P-value is equal to this value: P = 0.69, suggesting weak evidence against the null hypothesis.
03

Determine P-value for Lower-tailed test, n=20, t=-2.1

When n=20, this implies df=20-1=19. Given df = 19 and t = -2.1, we have a one-tailed p-value of around 0.025 according to the t-distribution table. Since this is a lower-tailed (or one-tailed) test, the P-value is the same as the one-tailed p-value: P = 0.025, suggesting moderate evidence against the null hypothesis.
04

Determine P-value for Lower-tailed test, n=20, t=-5.1

Given df = 19 (from n=20) and t = -5.1, we find a one-tailed p-value close to 0.001 (or possibly even less) in the t-distribution table. Again, because it's a lower-tailed test, the P-value equals the one-tailed p-value: P ≈ 0.001, suggesting strong evidence against the null hypothesis.
05

Determine P-value for Two-tailed test, n=40, t=1.7

Given df = 40-1=39 and t = 1.7, we find a one-tailed p-value of about 0.048 in the t-distribution table. Because this is a two-tailed test, we must double this value to get the P-value: 0.048*2 = 0.096, indicating weak to moderate evidence against 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 Vaia!

Key Concepts

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

t-distribution
The t-distribution is a key statistical distribution used in hypothesis testing, especially when dealing with small sample sizes. It's a kind of bell-shaped curve, much like the normal distribution, but it differs in that it has heavier tails.
This means the t-distribution is more forgiving to outliers in data, making it more appropriate for smaller datasets where sample sizes are limited. As the sample size increases, the t-distribution becomes more like the normal distribution.
  • Origin of the t-distribution: It was introduced by William Sealy Gosset for small sample datasets, who published under the pseudonym "Student," hence the name Student's t-distribution.
  • Application: Commonly used for estimating the mean in smaller sample sizes or when the standard deviation is unknown.
Whenever conducting a t-test, you will use the t-distribution tables to find probabilities associated with various t-scores and degrees of freedom.
degrees of freedom
Degrees of freedom (df) are a crucial consideration in statistical analyses, especially when using the t-distribution. In the context of the t-test, degrees of freedom determine the exact shape of the t-distribution.

To calculate the degrees of freedom for a t-test, use the formula: \[ df = n - 1 \] where "n" is the sample size.
  • Importance: Degrees of freedom impact the reliability of statistical tests and influence the decision threshold for statistical significance.
  • Effect: More degrees of freedom mean the t-distribution will have thinner tails and be closer to a normal distribution.
The degrees of freedom adjust as sample sizes increase, reflecting higher data reliability. When aligning sample statistics with population parameters, df plays an essential role in accurate assessments.
two-tailed test
A two-tailed test is used in hypothesis testing when we are interested in checking for effects in both directions, either an increase or a decrease.

Here's how it works:
  • Symmetric interest: If the null hypothesis states that a parameter has a specific value, a two-tailed test checks for any deviation, whether higher or lower.
  • Example: If we are testing if a mean is `10`, the two-tailed test will check if it's significantly different from `10`, whether it is larger or smaller.
  • P-value calculation: The p-value in a two-tailed test is calculated by doubling the one-tailed p-value because it considers the probability of deviation in both directions.
Hence, it is vital when both outcomes matter in statistical conclusions, ensuring that extremes on either side of the distribution contribute to rejecting the null hypothesis.
lower-tailed test
A lower-tailed test, also known as a left-tailed test, investigates if a sample statistic is significantly less than a stated value in the null hypothesis. This is useful when the primary concern is a reduction or minimization effect.

Key aspects of a lower-tailed test include:
  • Direction: It focuses on statistically significant lower values. We are testing if the observed statistic falls significantly below the hypothesized parameter.
  • P-value: The p-value is calculated as the area under the curve to the left of the calculated test statistic in a t-distribution.
  • Application: Useful for testing hypotheses where the interest is whether the effect might be less rather than greater than expected, such as testing if a new drug decreases blood pressure.
This type of test is one-sided, meaning it only values deviations in one specific direction, which can increase the test's power in detecting decreases.
upper-tailed test
An upper-tailed test is designed to determine if a sample statistic is significantly higher than a value specified in the null hypothesis. It is particularly useful in cases where a positive increase is the focus.

Here are some key features:
  • Focus: The test aims at identifying cases where the observed statistic is likely higher than the hypothesized statistic.
  • P-value calculation: P-values are obtained by looking at the right-hand tail of the distribution, indicating higher values.
  • Common Usage: If you are testing whether a new training method increases employees' productivity over a standard method, the interest is in effects on the upper end of the distribution.
Like a lower-tailed test, this one-sided test increases sensitivity when checking for potential increases, aiding in pinpoint precision for hypothesis validation.

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

The paper "MRI Evaluation of the Contralateral Breast in Women with Recently Diagnosed Breast Cancer" (New England Journal of Medicine \([2007]: 1295-1303)\) describes a study of the use of MRI (Magnetic Resonance Imaging) exams in the diagnosis of breast cancer. The purpose of the study was to determine if MRI exams do a better job than mammograms of determining if women who have recently been diagnosed with cancer in one breast have cancer in the other breast. The study participants were 969 women who had been diagnosed with cancer in one breast and for whom a mammogram did not detect cancer in the other breast. These women had an MRI exam of the other breast, and 121 of those exams indicated possible cancer. After undergoing biopsies, it was determined that 30 of the 121 did in fact have cancer in the other breast, whereas 91 did not. The women were all followed for one year, and three of the women for whom the MRI exam did not indicate cancer in the other breast were subsequently diagnosed with cancer that the MRI did not detect. The accompanying table summarizes this information. Suppose that for women recently diagnosed with cancer in only one breast, the MRI is used to decide between the two "hypotheses" \(H_{0}\) : woman has cancer in the other breast \(H_{a}:\) woman does not have cancer in the other breast (Although these are not hypotheses about a population characteristic, this exercise illustrates the definitions of Type I and Type II errors.) a. One possible error would be deciding that a woman who does have cancer in the other breast is cancerfree. Is this a Type I or a Type II error? Use the information in the table to approximate the probability of this type of error. b. There is a second type of error that is possible in this setting. Describe this error and use the information in the given table to approximate the probability of this type of error.

According to a Washington Post-ABC News poll, 331 of 502 randomly selected U.S. adults interviewed said they would not be bothered if the National Security Agency collected records of personal telephone calls they had made. Is there sufficient evidence to conclude that a majority of U.S. adults feel this way? Test the appropriate hypotheses using a .01 significance level.

In a survey conducted by CareerBuilder.com, employers were asked if they had ever sent an employee home because they were dressed inappropriately (June 17 . 2008 , www.careerbuilder.com). A total of 2765 employers responded to the survey, with 968 saying that they had sent an employee home for inappropriate attire. In a press release, CareerBuilder makes the claim that more than one- third of employers have sent an employee home to change clothes. Do the sample data provide convincing evidence in support of this claim? Test the relevant hypotheses using \(\alpha=.05 .\) For purposes of this exercise, assume that it is reasonable to regard the sample as representative of employers in the United States.

Assuming a random sample from a large population, for which of the following null hypotheses and sample sizes \(n\) is the large-sample \(z\) test appropriate: a. \(H_{0}: p=.2, n=25\) b. \(H_{0}: p=.6, n=210\) c. \(H_{0}: p=.9, n=100\) d. \(H_{0}: p=.05, n=75\)

A researcher speculates that because of differences in diet, Japanese children may have a lower mean blood cholesterol level than U.S. children do. Suppose that the mean level for U.S. children is known to be 170 . Let \(\mu\) represent the mean blood cholesterol level for all Japanese children. What hypotheses should the researcher test?

See all solutions

Recommended explanations on Math Textbooks

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.

Sign-up for free