91Ó°ÊÓ

The article "More Students Taking AP Tests" (San Luis Obispo Tribune, January 10,2003 ) provided the following information on the percentage of students in grades 11 and 12 taking one or more AP exams and the percentage of exams that earned credit in 1997 and 2002 for seven high schools on the central coast of California. $$ \begin{array}{cccccc} & {\begin{array}{c} \text { Percentage of } \\ \text { Students Taking } \\ \text { One or More } \\ \text { AP Exams } \end{array}} & & {\begin{array}{c} \text { Percentage of } \\ \text { Exams That } \\ \text { Earned College } \end{array}} \\ { 2 - 3 } { 5 - 6 } & & & & {\text { Credit }} \\ { 2 - 3 } { 5 - 6 } \text { School } & 1997 & 2002 & & 1997 & 2002 \\ & 1 & 13.6 & 18.4 & & 61.4 & 52.8 \\ 2 & 20.7 & 25.9 & & 65.3 & 74.5 \\ 3 & 8.9 & 13.7 & & 65.1 & 72.4 \\ 4 & 17.2 & 22.4 & & 65.9 & 61.9 \\ 5 & 18.3 & 43.5 & & 42.3 & 62.7 \\ 6 & 9.8 & 11.4 & & 60.4 & 53.5 \\ 7 & 15.7 & 17.2 & & 42.9 & 62.2 \\ \hline \end{array} $$ a. Assuming that it is reasonable to regard these seven schools as a random sample of high schools located on the central coast of California, carry out an appropriate test to determine if there is convincing evidence that the mean percentage of exams earning college credit at central coast high schools in 1997 and in 2002 were different. b. Do you think it is reasonable to generalize the conclusion of the test in Part (a) to all California high schools? Explain. c. Would it be appropriate to use the paired-samples \(t\) test with the data on percentage of students taking one or more AP exams? Explain.

Step by step solution

01

Preparing the Data

Organize the data from the percentage of exams that earned college credit for the years 1997 and 2002 into two different arrays or lists.

02

Calculating the Means

Calculate the mean percentage of exams that earned college credit for both 1997 and 2002.

03

Applying the Formula for the t-test

Calculate the standard deviation and standard error for each sample (1997 and 2002). Then use these values to conduct a two-sample t-test. The formula for the t-score is \( t = \frac{\bar{x_{1}} - \bar{x_{2}} }{\sqrt{ s_{1}^{2}/n_{1} + s_{2}^{2}/n_{2} }} \) where \( \bar{x_{1}} \) and \( \bar{x_{2}} \) are the sample means, \( s_{1}^{2} \) and \( s_{2}^{2} \) are the sample variances and \( n_{1} \) and \( n_{2} \) are the sample sizes.

04

Understanding the Result

Determine whether there is significant difference between the means of the two groups by comparing the obtained t value with the critical t value for a given level of significance (typically 0.05). If the obtained t value is greater than the critical t value, then there is a significant difference between the groups.

05

Generalizing the Results

Think critically about whether these results can be generalized for all high schools in California, keeping in mind representativeness of the sample, sample size, and possible biases.

06

Evaluating the Appropriateness of t-test

Determine whether it would be appropriate to use paired-samples t-test with the data on percentage of students taking one or more AP exams. A paired-samples t-test would require each observation in one group to correspond with an observation in the second group. Since the data provided in the exercise doesn't appear to be based on paired observations, paired t-test may not be appropriate.

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Key Concepts

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

Two-Sample t-Test

When investigating the average differences between two independent groups, statisticians often employ the two-sample t-test. This statistical method compares two sample means to see whether there is enough evidence to suggest that the means of the corresponding populations are significantly different from each other.

The two-sample t-test is based on the assumption that the two groups are independent, have normally distributed data, and have similar variances. The formula to calculate the t-score is: \[ t = \frac{\bar{x_{1}} - \bar{x_{2}} }{\sqrt{ s_{1}^{2}/n_{1} + s_{2}^{2}/n_{2} }} \]where \( \bar{x_{1}} \) and \( \bar{x_{2}} \) are the sample means for each group, \( s_{1}^{2} \) and \( s_{2}^{2} \) are the variances, and \( n_{1} \) and \( n_{2} \) are the sample sizes.This calculation generates a t-score, which is then compared to a critical value from a t-distribution table. If the t-score exceeds the critical value at a defined level of significance (often 0.05), we conclude there's a significant difference between the groups.
In our case, the aim is to compare the mean percentage of exams that earned college credit at central coast high schools in 1997 and 2002. By applying the two-sample t-test to our AP exam data, we can statistically infer whether the performance in these two years was significantly different.

Sample Means Calculation

The sample mean is a critical value in many statistical analyses, including t-tests, as it serves as an estimate of the population mean. Calculating the sample mean is straightforward: sum all the measurements in the sample and then divide by the number of observations in the sample. The mathematical notation for the sample mean is \( \bar{x} \), and the calculation can be expressed as: \[ \bar{x} = \frac{1}{n}\sum_{i=1}^{n}x_{i} \]where \( x_i \) represents each value in the sample and \( n \) represents the sample size.When analyzing AP exam data, as is the case in our example, calculating the sample mean for each year's percentage of exams that earned college credit gives us a clear idea of the average performance for that year. Understanding and accurately calculating this value is essential, as it's the cornerstone of the subsequent statistical test—the two-sample t-test.

Data Representativeness

Data representativeness is concerned with how accurately a sample reflects the population from which it's drawn. For the conclusions drawn from statistical tests to be valid for the broader population, the sample must be representative. It involves ensuring that a variety of factors, such as demographics or geographic location, which might influence the variables being studied, are proportionally included in the sample.

When considering AP exam data, we must engage with questions of representativeness. For instance, when we look at AP exam results from seven high schools on the central coast of California, is it fair to generalize our findings to all California high schools? It's important to acknowledge that schools across the state might differ in significant ways, such as in resources, teaching quality, or student socio-economic backgrounds.
All these aspects might impact AP exam outcomes, which might mean our sample's findings wouldn't hold true for every school. Because representativeness affects the validity of our conclusions, it must be carefully considered, and any potential limitations should be transparently discussed.