91Ó°ÊÓ

Large Data Set 1 lists the GPAs of 1,000 students. a. Regard the data as arising from a census of all freshman at a small college at the end of their first academic year of college study, in which the GPA of every such person 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 is less than \(2.50,\) at the \(10 \%\) level of significance. (The null hypothesis is that \(\mu=2.50 .)\) c. Is your conclusion in part \((b)\) in agreement with the true state of nature (which by part (a) you 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 Population Mean (Part a)

To find the population mean \( \mu \), sum all GPAs from Large Data Set 1 and divide by 1,000 (the total number of students). Let \( X_1, X_2, \ldots, X_{1000} \) represent the GPAs of the students, then the population mean \( \mu \) is computed as:\[\mu = \frac{X_1 + X_2 + \cdots + X_{1000}}{1000}.\]

02

Set Up Hypothesis Test (Part b)

We are testing the hypothesis that the population mean is less than 2.50. This means:- Null Hypothesis \( H_0: \mu = 2.50 \)- Alternative Hypothesis \( H_a: \mu < 2.50 \).

03

Calculate Sample Statistics

From the first 50 GPAs (the sample), calculate the sample mean \( \bar{x} \) and standard deviation \( s \). Let these be denoted as \( X_1, X_2, \ldots, X_{50} \). Compute:\[\bar{x} = \frac{X_1 + X_2 + \cdots + X_{50}}{50}\]and estimate the sample standard deviation \( s \) using the formula:\[s = \sqrt{\frac{\sum_{i=1}^{50} (X_i - \bar{x})^2}{49}}\]

04

Determine Test Statistic and Critical Value

Calculate the test statistic using the formula for a one-sample t-test:\[t = \frac{\bar{x} - 2.50}{s / \sqrt{50}}.\]Determine the critical value for \( t \) at \( 49 \) degrees of freedom at the 10% significance level for a one-tailed test using a t-distribution table.

05

Make the Decision

Compare the test statistic with the critical value:- If the test statistic is less than the critical value, reject the null hypothesis \( H_0 \).- Otherwise, do not reject \( H_0 \).

06

Evaluate Conclusion (Part c)

Based on the decision in Step 5, compare to the actual population mean from Step 1:- If you rejected \( H_0 \) and \( \mu \) is really less than 2.50, your decision is correct.- If you did not reject \( H_0 \) but \( \mu \) is less than 2.50, your decision is a Type II error.- If you rejected \( H_0 \) and \( \mu \) actually equals 2.50, your decision is a Type I error.

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Key Concepts

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

Understanding the Population Mean

The population mean is the true average of a particular characteristic in a complete population. In our exercise, the population consists of 1,000 students and their GPAs. To find the population mean \(\mu\), we sum all of the GPAs from this group and then divide by 1,000.

This value gives us a clearer picture of the general performance level of all freshmen at the end of their first year. Since this figure encompasses every individual in the population, it is not an estimate but an exact measurement of average GPA at that school.

Exploring the Sample Mean

When we don't have access to data for an entire population, we turn to a sample mean as an estimate. In our use case, the first 50 students from the 1,000 are treated as a sample. The sample mean \(\bar{x}\) is calculated by adding together the GPAs of these 50 students and then dividing by 50.

This sample mean serves as a proxy to help approximate the population mean. However, the sample mean is always an estimate and can vary depending on which group of students are selected.

Type I and Type II Errors Explained

When conducting a hypothesis test, errors can occur. A Type I error happens when a true null hypothesis is rejected. In the exercise, if we conclude that the population mean GPA is less than 2.50, but in reality, it is equal to 2.50, we've made a Type I error.

Conversely, a Type II error occurs when we fail to reject a false null hypothesis. This would mean we conclude the GPA is at or above 2.50, while in fact it is below 2.50. Understanding these errors is crucial since they influence the confidence level and potential consequences of the test results.

Diving into Critical Value and Test Statistic

In hypothesis testing, the test statistic determines if our sample data supports rejecting the null hypothesis. It's calculated using the sample mean, assumed population mean under null hypothesis, sample standard deviation, and sample size.

The critical value is derived from a statistical distribution (like t-distribution) at a specific significance level. In this context, it's the threshold that the test statistic is compared to. If the test statistic is more extreme than the critical value, the null hypothesis is rejected.

This comparison tells us whether there is enough evidence in our sample to infer that the population parameter is different from the hypothesized value.