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Chapter 10: Problem 132

Use the following information to answer the next ten exercises. indicate which of the following choices best identifies the hypothesis test. a. independent group means, population standard deviations and/or variances known b. independent group means, population standard deviations and/or variances unknown c. matched or paired samples d. single mean e. two proportions f. single proportion A new SAT study course is tested on 12 individuals. Pre-course and post-course scores are recorded. Of interest is the mean increase in SAT scores. The following data are collected: table cannot copy

Short Answer

Expert verified
The hypothesis test type is matched or paired samples (choice c).

Step by step solution

01

Identify the Objective

The objective is to determine the type of hypothesis test to be used based on the scenario provided. The focus is on the mean increase in SAT scores for the same set of individuals before and after a course.
02

Analyze the Data Collection Method

The data involves pre-course and post-course SAT scores for the same 12 individuals, indicating that these are repeated measures on the same subjects. This setup suggests a relationship between each pair of scores, as they are not from independent groups.
03

Identify Test Type Using Data Characteristics

Since the data involves pairs of measurements for the same individuals and we are interested in the mean increase in scores, we are dealing with matched or paired data. The choice should be based on the definition of hypotheses tests involving paired samples, where dependencies between paired observations are considered.

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Key Concepts

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

Paired Samples
When you are dealing with paired samples, you are essentially focusing on two groups of data that are somehow connected or related. In hypothesis testing, this typically arises when the same subjects are measured in two different scenarios, like before and after a treatment or intervention.
This relationship between data points means we're not dealing with independent groups but rather observations that are linked.
  • Each subject provides data for both samples.
  • Common in "before and after" studies.
  • The test considers the differences between paired observations.
Recognizing paired samples is crucial because the method of analysis will differ from those used for independent groups. This ensures that the underlying dependencies in the data are acknowledged, allowing for more accurate statistical conclusions.
Mean Increase
The term "mean increase" refers to the average change in a variable of interest, often observed before and after an intervention. In our context, it鈥檚 about determining whether the SAT scores increased by an average amount after taking a study course.
For paired sample data like ours, calculating the mean increase involves finding the difference between each individual's post-course and pre-course scores.
  • First, calculate the difference for each pair of observations.
  • Then, compute the average of these differences.
This average difference tells us the mean increase (or decrease) in scores. Not only does this quantify the change, but it also forms the basis for testing if this change is statistically significant.
Repeated Measures
Repeated measures involve collecting data from the same subjects at multiple time points. It helps in analyzing how a particular variable changes over time or under different conditions.
This is exactly what happens in our SAT study, where each participant is measured twice - once before the course, and once after.
  • This method allows for observing individual changes.
  • Very useful for tracking developments over time.
  • It reduces variability because the same subjects are involved in each measurement.
Repeated measures ensure that personal characteristics of subjects, which might affect the outcome, are naturally controlled. This makes conclusions drawn from such analyses more robust and personalized.
SAT Scores
SAT scores are standardized test results that are often used in college admissions in the United States. When analyzing SAT scores in studies, especially with the goal of evaluating educational interventions, it's crucial to understand how they are measured.
In the context of our problem, SAT scores serve as the indicator of academic achievement before and after the course.
  • Changes in these scores reflect the impact of a preparatory course.
  • They offer a concrete measure to assess the effectiveness of the intervention.
Using SAT scores as a metric provides educators and researchers with tangible evidence of learning gains or needs, guiding decisions about educational strategies and resource allocations.

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Most popular questions from this chapter

Use the following information to answer the next 12 exercises: The U.S. Center for Disease Control reports that the mean life expectancy was 47.6 years for whites born in 1900 and 33.0 years for nonwhites. Suppose that you randomly survey death records for people born in 1900 in a certain county. Of the 124 whites, the mean life span was 45.3 years with a standard deviation of 12.7 years. Of the 82 nonwhites, the mean life span was 34.1 years with a standard deviation of 15.6 years. Conduct a hypothesis test to see if the mean life spans in the county were the same for whites and nonwhites. Sketch a graph of the situation. Label the horizontal axis. Mark the hypothesized difference and the sample difference. Shade the area corresponding to the p-value.

Use the following information to answer the next 12 exercises: The U.S. Center for Disease Control reports that the mean life expectancy was 47.6 years for whites born in 1900 and 33.0 years for nonwhites. Suppose that you randomly survey death records for people born in 1900 in a certain county. Of the 124 whites, the mean life span was 45.3 years with a standard deviation of 12.7 years. Of the 82 nonwhites, the mean life span was 34.1 years with a standard deviation of 15.6 years. Conduct a hypothesis test to see if the mean life spans in the county were the same for whites and nonwhites. At a pre-conceived \(\alpha=0.05,\) what is your: a. Decision: b. Reason for the decision: c. Conclusion (write out in a complete sentence):

Use the following information to answer the next 15 exercises: Indicate if the hypothesis test is for a. independent group means, population standard deviations, and/or variances known b. independent group means, population standard deviations, and/or variances unknown c. matched or paired samples d. single mean e. two proportions f. single proportion In a random sample of 100 forests in the United States, 56 were coniferous or contained conifers. In a random sample of 80 forests in Mexico, 40 were coniferous or contained conifers. Is the proportion of conifers in the United States statistically more than the proportion of conifers in Mexico?

Use the following information to answer the next five exercises. A researcher is testing the effects of plant food on plant growth. Nine plants have been given the plant food. Another nine plants have not been given the plant food. The heights of the plants are recorded after eight weeks. The populations have normal distributions. The following table is the result. The researcher thinks the food makes the plants grow taller. $$\begin{array}{|l|l|l|}\hline \text { Plant Group } & {\text { Sample Mean Height of Plants (inches) }} & {\text { Population Standard Deviation }} \\\ \hline \text { Food } & {16} & {2.5} \\ \hline \text { No food } & {14} & {1.5} \\ \hline\end{array}$$ Is the population standard deviation known or unknown?

Use the following information to answer the next two exercises. A new AIDS prevention drug was tried on a group of 224 HIV positive patients. Forty-five patients developed AIDS after four years. In a control group of 224 HIV positive patients, 68 developed AIDS after four years. We want to test whether the method of treatment reduces the proportion of patients that develop AIDS after four years or if the proportions of the treated group and the untreated group stay the same. Let the subscript t = treated patient and ut = untreated patient. If the \(p\) -value is 0.0062 what is the conclusion (use \(\alpha=0.05 ) ?\) a. The method has no effect. b. There is sufficient evidence to conclude that the method reduces the proportion of HIV positive patients who develop AIDS after four years. c. There is sufficient evidence to conclude that the method increases the proportion of HIV positive patients who develop AIDS after four years. d. There is insufficient evidence to conclude that the method reduces the proportion of HIV positive patients who develop AIDS after four years.
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