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Chapter 9: Problem 82

Choose a \(t\) -test for each situation: one-sample \(t\) -test, twosample \(t\) -test, paired \(t\) -test, and no \(t\) -test. a. A random sample of car dealerships is obtained. Then a student walks onto each dealer's lot wearing old clothes and finds out how long it takes (in seconds) for a salesperson to approach the student. Later the student goes onto the same lot dressed very nicely and finds out how long it takes for a salesperson to approach. b. A researcher at a preschool selects a random sample of 4 -year-olds, determines whether they know the alphabet (yes or no), and records gender. c. A researcher calls the office phone for a random sample of faculty at a college late at night, measures the length of the outgoing message, and records gender.

Short Answer

Expert verified
For situation a, a paired t-test is used. For situation b, no t-test is appropriate as the outcome is categorical. For situation c, a two-sample t-test is suitable.

Step by step solution

01

Apply the correct t-test for situation a

In scenario A, the same individual is measured twice under different conditions. We have pairs of observations. Hence, a paired t-test is appropriate.
02

Decide the correct t-test for situation b

In scenario B, the researcher is testing two independent groups (gender) and is noting a categorical outcome (knowing the alphabet). Hence, t-test is not appropriate.
03

Select the appropriate t-test for situation c

In scenario C, the researcher is testing two independent groups (gender) for a continuous outcome (length of the outgoing message). Hence, a two-sample t-test is correct.

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Key Concepts

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

Paired t-test
When dealing with experiments or observations that involve two related groups, the paired t-test is your go-to method. This statistical test compares two dependent samples, typically to find if there's a significant difference before and after a specific treatment or intervention.

Think of it as evaluating the impact of a change under controlled conditions on the same subjects. For example, if you want to test a new teaching method's effectiveness, you'd measure student performance before and after applying the method.

Real-world Application

Consider a weight loss program where participants' weights are recorded before and after the completion of the program. Here, each participant serves as their control, and the paired t-test helps determine if the program significantly affected their weight.
Two-sample t-test
The two-sample t-test, also known as the independent samples t-test, is a staple in comparing means from two different groups. This test assumes that the two groups are independent and unrelated.

The primary aim is to determine whether the means of two independent samples are significantly different from one another. A classic example would be testing whether the average height of men differs from that of women in a random population sample.

Simplifying Complex Analyses

A study comparing the effects of two different teaching methods on two different groups of students would utilize a two-sample t-test to assess the efficiency of these methods.
Statistical hypothesis testing
Statistical hypothesis testing is a core concept in inferential statistics used to determine the likelihood that a result observed within a data sample can occur by chance.

The process usually starts with a null hypothesis, which assumes no effect or no difference between groups, and an alternative hypothesis, which suggests a significant effect or difference. A p-value determined during the test will dictate whether we can reject the null hypothesis in favor of the alternative hypothesis.

Conceptual Foundation

In essence, hypothesis testing acts as the judge in the 'trial' of whether an observed data pattern is a genuine effect or just a coincidence. It is the foundation of various statistical tests, including the t-tests.
Categorical outcome
A categorical outcome refers to a variable that can take on one of a limited, and usually fixed, number of possible values, assigning each individual to a particular group or nominal category.

Such outcomes are common in research where you might be interested in 'Yes' or 'No' responses, such as whether a customer would purchase a product again or not. Unlike continuous outcomes that may be measured by means and require tests like the t-test, categorical outcomes are typically analyzed using chi-square or Fisher's exact tests.

Understanding Through Examples

To illustrate, analyzing survey results where respondents are asked to choose their favorite type of music would involve dealing with categorical data, as music genres are distinct categories without inherent order.

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Most popular questions from this chapter

Women's Heights Assume women's heights are approximately Normally distributed with a mean of 65 inches and a standard deviation of \(2.5\) inches. Which of the following questions can be answered using the Central Limit Theorem for sample means as needed? If the question can be answered, do so. If the question cannot be answered, explain why the Central Limit Theorem cannot be applied. a. Find the probability that a randomly selected woman is less than 63 inches tall. b. If five women are randomly selected, find the probability that the mean height of the sample is less than 63 inches. c. If 30 women are randomly selected, find the probability that the mean height of the sample is less than 63 inches.

Potatoes The weights of four randomly and independently selected bags of potatoes labeled 20 pounds were found to be \(21.0\), \(21.5,20.5\), and \(21.2\) pounds. Assume Normality. a. Find a \(95 \%\) confidence interval for the mean weight of all bags of potatoes. b. Does the interval capture \(20.0\) pounds? Is there enough evidence to reject a mean weight of 20 pounds?

Why Is \(n-1\) in the Sample Standard Deviation? Why do we calculate \(s\) by dividing by \(n-1\), rather than just \(n\) ? $$ s^{2}=\frac{\sum(x-\bar{x})^{2}}{n-1} $$ The reason is that if we divide by \(n-1\), then \(s^{2}\) is an unbiased estimator of \(\sigma^{2}\), the population variance. We want to show that \(s^{2}\) is an unbiased estimator of \(\sigma^{2}\), sigma squared. The mathematical proof that this is true is beyond the scope of an introductory statistics course, but we can use an example to demonstrate that it is. First we will use a very small population that consists only of these three numbers: 1,2, and 5 . You can determine that the population standard deviation, \(\sigma\), for this population is \(1.699673\) (or about \(1.70\) ), as shown in the \(\mathrm{TI}-84\) output. So the population variance, sigma squared, \(\sigma^{2}\), is \(2.888889\) (or about 2.89). Now take all possible samples, with replacement, of size 2 from the population, and find the sample variance, \(s^{2}\), for each sample. This process is started for you in the table. Average these sample variances \(\left(s^{2}\right)\), and you should get approximately \(2.88889 .\) If you do. then you have demonstrated that \(s^{2}\) is an unbiased estimator of \(\sigma^{2}\), sigma squared. Show your work by filling in the accompanying table and show the average of \(s^{2}\).

Choose a test for each situation: one-sample \(t\) -test, two-sample \(t\) -test, paired \(t\) -test, and no \(t\) -test. a, A random sample of students who transfered to a 4 -year university from community colleges are asked their GPAs. Our goal is to determine whether the mean GPA for transfer students is significantly different from the population mean GPA for all students at the university. b. Students observe the number of office hours posted for a random sample of tenured and a random sample of untenured professors. c. A researcher goes to the parking lot at a large grocery chain and observes whether each person is male or female and whether they return the cart to the correct spot before leaving (yes or no).

Exam Scores The distribution of the scores on a certain exam is \(N(100,10)\) which means that the exam scores are Normally distributed with a mean of 100 and a standard deviation of \(10 .\) a. Sketch or use technology to create the curve and label on the \(x\) -axis the position of the mean, the mean plus or minus one standard deviation, the mean plus or minus two standard deviations, and the mean plus or minus three standard deviations. b. Find the probability that a randomly selected score is between 90 and 110 . Shade the region under the Normal curve whose area corresponds to this probability.
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