91Ó°ÊÓ

Large Data Set 1 lists the SAT scores of 1,000 students. a. Regard the data as arising from a census of all students at a high school, in which the SAT score of every student was measured. Compute the population mean \(\mu\). b. Regard the first 50 students in the data set as a random sample drawn from the population of part (a) and use it to test the hypothesis that the population mean exceeds \(1,510,\) at the \(10 \%\) level of significance. (The null hypothesis is that \(\mu=1510 .\) ) (a) you c. Is your conclusion in part (b) in agreement with the true state of nature (which by part know), or is your decision in error? If your decision is in error, is it a Type I error or a Type II error?

Step by step solution

01

Calculate the Population Mean

To find the population mean \( \mu \), sum all the SAT scores of the 1,000 students and divide by 1,000. This can be denoted as:\[ \mu = \frac{1}{1000} \sum_{i=1}^{1000} x_i \] where \( x_i \) represents the score of each student.

02

Define the Hypotheses for the Sample Test

For part (b), define the null hypothesis \( H_0: \mu = 1510 \) and the alternative hypothesis \( H_a: \mu > 1510 \). We are testing if the population mean exceeds 1510 using the first 50 student scores as a sample.

03

Calculate the Sample Mean and Standard Deviation

Calculate the sample mean \( \bar{x} \) and the sample standard deviation \( s \) for the 50 students' scores. Use the formula:\[ \bar{x} = \frac{1}{50} \sum_{i=1}^{50} x_i \] and the sample standard deviation:\[ s = \sqrt{\frac{1}{49} \sum_{i=1}^{50} (x_i - \bar{x})^2} \]

04

Conduct a One-Sample t-Test

Perform a one-sample t-test using the sample mean, standard deviation, and sample size (n=50). Compute the t-statistic:\[ t = \frac{\bar{x} - 1510}{s/\sqrt{50}} \] Compare the t-statistic to the critical t-value from the t-distribution table at \( \alpha = 0.10 \) with 49 degrees of freedom.

05

Make a Decision Based on t-Test Results

If the calculated t-statistic is greater than the critical t-value, reject the null hypothesis \( H_0 \); otherwise, fail to reject it. This decision indicates whether there is enough evidence to conclude the population mean exceeds 1510.

06

Verify Conclusion with Known Population Mean

Compare the conclusion from the t-test to the known population mean (from part a). If the decision from the hypothesis test matches that the true mean is not greater than 1510, it is correct; otherwise, it's an error.

07

Determine the Type of Error (if any)

If the decision in Step 5 is incorrect, classify the error. - A Type I error occurs if we wrongly rejected a true null hypothesis. - A Type II error occurs if we failed to reject a false null hypothesis.

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Key Concepts

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

Population Mean

The population mean, denoted by \( \mu \), is a key concept in statistics. It represents the average of a set of values for an entire population. In the context of SAT scores, we regard all scores of the 1,000 students in the data set to compute the population mean. To calculate this, sum all individual SAT scores and divide the total by 1,000 since there are 1,000 students. This can be shown mathematically as:

• \( \mu = \frac{1}{1000} \sum_{i=1}^{1000} x_i \)
• Each \( x_i \) is an individual student's score.

The calculated result gives an estimate of the central tendency of SAT scores for the entire student population.

Sample Mean

The sample mean, denoted as \( \bar{x} \), is similar to the population mean but is calculated from a smaller subset of the population. In hypothesis testing, we often rely on sample data (such as the first 50 students' scores) to make inferences about the population mean. You compute the sample mean by adding up all the scores in your sample and dividing by the number of scores, typically:

• \( \bar{x} = \frac{1}{50} \sum_{i=1}^{50} x_i \)
• \( x_i \) represents each student's score in the sample.

This gives an average score for the sample, which serves as an estimate to infer information about the entire population's performance.

T-Test

A t-test is a statistical method used to determine if there is a significant difference between the means of two groups or between a sample mean and a known population mean. In our case, we perform a one-sample t-test to examine whether the sample mean of the SAT scores is significantly greater than 1510. The process is as follows:

• State the null hypothesis \( H_0 : \mu = 1510 \) and the alternative hypothesis \( H_a : \mu > 1510 \).
• Calculate the sample mean \( \bar{x} \), standard deviation \( s \), and then the t-statistic.
• \( t = \frac{\bar{x} - 1510}{s/\sqrt{50}} \)
• Compare the t-statistic with the critical t-value from the t-distribution table at a \( 10\% \) significance level.

If the calculated t-statistic exceeds the critical value, we reject the null hypothesis and conclude that the population mean is potentially greater than 1510.

Type I and Type II Errors

Errors in hypothesis testing occur when a decision made based on a statistical test doesn't correspond to the true state of a population. Type I and Type II errors are crucial concepts here:

• Type I Error: This happens when we reject a true null hypothesis. It means that based on the sample data, we wrongly conclude that \( \mu > 1510 \) when, in fact, it is equal to 1510 or less.
• Type II Error: This arises when we fail to reject a false null hypothesis. In this case, we might conclude that \( \mu \) is equal to or less than 1510 when it is actually greater.

To minimize these errors, chose an appropriate significance level, usually balancing between sensitivity (Type II) and specificity (Type I) within your testing framework.