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A researcher wants to know if students who do not plan to apply for financial aid had more variability on the SAT I math test than those who plan to do so. She obtains a random sample of 35 students who do not plan to apply for financial aid and a random sample of 38 students who do plan to apply for financial aid and obtains the following results: $$ \begin{array}{cc} \begin{array}{l} \text { Do Not Plan to Apply } \\ \text { for Financial Aid } \end{array} & \begin{array}{l} \text { Plan to Apply } \\ \text { for Financial Aid } \end{array} \\ \hline n_{1}=35 & n_{2}=38 \\ \hline s_{1}=123.1 & s_{2}=119.4 \\ \hline \end{array} $$ Do students who do not plan to apply for financial aid have a higher standard deviation on the SAT I math exam than students who plan to apply for financial aid at the \(\alpha=0.01\) level of significance? SAT I math exam scores are known to be normally distributed.

Step by step solution

01

Define the Hypotheses

First, state the null and alternative hypotheses. The null hypothesis (H鈧�) assumes that the variances are equal, and the alternative hypothesis (H鈧�) assumes that the variance of the scores for those who do not plan to apply for financial aid is greater than for those who do plan to apply.\[ H鈧�: \ \ \sigma^2_1 = \sigma^2_2 \ \ H鈧�: \ \ \sigma^2_1 > \sigma^2_2 \]

02

Identify the Test Statistic

Use the F-test for comparing two variances. The test statistic is defined as: \[ F = \frac{ s^2_1 }{ s^2_2 } \]Where \( s^2_1 \) and \( s^2_2 \) are the sample variances for the two groups.

03

Calculate the Test Statistic

Perform the calculation using the sample standard deviations provided: \[ s_1 = 123.1 \quad \ n_1 = 35 \ s_2 = 119.4 \quad \ n_2 = 38 \] \[ F = \frac{ (123.1)^2 }{ (119.4)^2 } = \frac{ 15148.41 }{ 14256.36 } \approx 1.063 \]

04

Determine the Critical Value

Look up the critical value for an F-distribution with \(df_1 = n_1 - 1 = 34\) and \(df_2 = n_2 - 1 = 37\) at the \( \alpha = 0.01 \) level of significance (one-tailed test).

05

Compare the Test Statistic to the Critical Value

Compare the calculated F-value with the critical F-value from statistical tables or software. If the calculated F-value is greater than the critical value, reject the null hypothesis.

06

Conclusion

Because the calculated F-value (1.063) is less than the critical F-value, we fail to reject the null hypothesis. Thus, there is no significant evidence to conclude that students who do not plan to apply for financial aid have a higher standard deviation in SAT I math scores than those who plan to apply for financial aid at the 0.01 significance level.

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Key Concepts

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

F-test

The F-test is a statistical test that is used to compare the variances of two populations. Variance measures the spread or variability within a set of data points. In simpler terms, it helps determine if two sets of data have different levels of dispersion. To perform an F-test, you start by calculating the ratio of the two sample variances. This ratio forms the F-test statistic. The formula for the F-test statistic is:

\[ F = \frac{s^2_1}{s^2_2} \]

Here, \( s^2_1 \) and \( s^2_2 \) are the sample variances. The calculated F-value is then compared to a critical value from the F-distribution table. If the F-value is larger than the critical value, we conclude that there is a significant difference between the variances of the two samples. The F-test is powerful because it helps to validate the variability differences between two groups and is commonly used in hypothesis testing scenarios.

Hypothesis Testing

Hypothesis testing is a statistical method for making decisions using data. It involves an initial assumption, called the null hypothesis (H鈧�), and an alternative hypothesis (H鈧�) which is what you want to prove. In our example, we're testing whether the variance for students who don't plan to apply for financial aid is greater than for those who do.

The null hypothesis assumes no difference between the variances, while the alternative suggests otherwise:

\[ H鈧� : \sigma^2_1 = \sigma^2_2 \]
\[ H鈧� : \sigma^2_1 > \sigma^2_2 \]
Once you have set up your hypotheses, you use statistical tests (like the F-test) to decide whether to reject the null hypothesis. If the test shows significant evidence against H鈧�, you reject it in favor of H鈧�. Otherwise, you fail to reject H鈧�, indicating insufficient evidence to support H鈧�.

Significance Level

The significance level, often denoted as \( \alpha \), is the probability threshold for rejecting the null hypothesis. It defines how confident you need to be in your results before considering them statistically significant. A common choice for \( \alpha \) is 0.05, but in our example, it is 0.01. A lower \( \alpha \) makes the test more stringent.

If the probability of obtaining the observed results under the null hypothesis is less than \( \alpha \), you reject the null hypothesis. This implies that the observed data is highly unlikely under H鈧�. Conversely, if the p-value (probability of obtaining the observed data) is greater than \( \alpha \), you fail to reject the null hypothesis. In simpler terms, \( \alpha \) helps control the likelihood of making a Type I error, which is incorrectly rejecting a true null hypothesis.

Standard Deviation

Standard deviation is a key measure of variability in a dataset. It tells you how much individual data points deviate from the mean (average). A higher standard deviation indicates that data points are spread out more, whereas a lower standard deviation means they are closer to the mean.

The formula for standard deviation is:

\[ s = \sqrt{ \frac{1}{N-1} \sum_{i=1}^N (x_i - \bar{x})^2 } \]

Here, \((x_i - \bar{x})\) represents the deviation of each data point from the mean, \( \bar{x}\). When comparing two groups, differences in their standard deviations can provide insights into their variability. For instance, if one group has a higher standard deviation than another, it suggests more dispersion in its data points.

Sample Variance

Sample variance measures the spread of data points in a sample and is the square of the standard deviation. It provides insights into how much the data varies around the mean. The formula for sample variance is:

\[ s^2 = \frac{1}{N-1} \sum_{i=1}^N (x_i - \bar{x})^2 \]

Here, \((x_i - \bar{x})\) represents the deviation of each data point from the mean, \( \bar{x}\). Sample variance is crucial when comparing two sets of data because it directly influences the F-test statistic. In the context of hypothesis testing, it helps determine whether differences in variability between two groups are significant or just due to random chance.