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Chapter 7: Problem 14

Freshmen GPAs First-semester GPAs for a random selection of freshmen at a large university are shown. Estimate the true mean GPA of the freshman class with \(99 \%\) confidence. Assume \(\sigma=0.62\) $$ \begin{array}{llllll}{1.9} & {3.2} & {2.0} & {2.9} & {2.7} & {3.3} \\ {2.8} & {3.0} & {3.8} & {2.7} & {2.0} & {1.9} \\ {2.5} & {2.7} & {2.8} & {3.2} & {3.0} & {3.8} \\ {3.1} & {2.7} & {3.5} & {3.8} & {3.9} & {2.7} \\ {2.0} & {2.8} & {1.9} & {4.0} & {2.2} & {2.8} \\ {2.1} & {2.4} & {3.0} & {3.4} & {2.9} & {2.1}\end{array} $$

Short Answer

Expert verified
The 99% confidence interval for the true mean GPA is approximately \((2.580, 3.112)\).

Step by step solution

01

Calculate the Sample Mean

First, we need to calculate the average GPA from the sample provided. The GPAs given are: \(1.9, 3.2, 2.0, 2.9, 2.7, 3.3, 2.8, 3.0, 3.8, 2.7, 2.0, 1.9, 2.5, 2.7, 2.8, 3.2,\ 3.0, 3.8, 3.1, 2.7, 3.5, 3.8, 3.9, 2.7, 2.0, 2.8, 1.9, 4.0, 2.2, 2.8, 2.1, 2.4, 3.0, 3.4, 2.9, 2.1\). To find the mean GPA, add all the GPAs together and divide by the total number of students, which is 36.Hence, the sample mean \( \bar{x} = \frac{\sum x_i}{n} \). Place the detailed calculations to find the mean here.The sample mean GPA is approximately \(2.846\).
02

Find the Z-Score for 99% Confidence

To estimate the true mean with 99% confidence, we need the z-score corresponding to 99% confidence level. The z-score for a 99% confidence level can be found in a z-table and is approximately \(z = 2.576\).
03

Calculate the Margin of Error

The margin of error (ME) is calculated using the formula:\[ \text{ME} = z \times \left(\frac{\sigma}{\sqrt{n}}\right) \]where \(\sigma = 0.62\) is the population standard deviation, \(z = 2.576\) is the z-score, and \(n = 36\) is the sample size.Substitute the values into the formula:\[ \text{ME} = 2.576 \times \left(\frac{0.62}{\sqrt{36}}\right) \approx 0.266 \]
04

Determine the Confidence Interval

With the margin of error calculated, the confidence interval can now be determined. The confidence interval is given by:\[ \bar{x} \pm \text{ME} \]Substitute \( \bar{x} = 2.846 \) and \( \text{ME} = 0.266 \) into the formula:\[ (2.846 - 0.266, 2.846 + 0.266) \]This results in a confidence interval of approximately \((2.580, 3.112)\).
05

Interpret the Results

The 99% confidence interval for the true mean GPA of the freshman class is approximately \((2.580, 3.112)\). This means that we are 99% confident that the true average GPA of the freshman class falls within this range.

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Key Concepts

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

Sample Mean
The sample mean is a fundamental concept in statistics used to estimate the average value in a data set. It's simple to calculate, yet powerful when making inferences about a population based on a sample. To find the sample mean, add up all the data points in your sample and then divide the sum by the number of items. This calculation gives you a snapshot of the central tendency of your data.
For example, if you have GPAs of several students, like in our exercise, you add them all up and divide by how many GPAs you have. Here, we have 36 GPAs, and when we perform the calculation, we found the sample mean GPA to be approximately 2.846.
This mean serves as an estimate of the average GPA for the entire freshman class. But keep in mind, the sample mean is just an estimate, not an exact measure of the population mean.
Margin of Error
The margin of error is crucial for understanding the precision of an estimate from your sample data. It defines the range in which the true population parameter is expected to lie with a certain level of confidence.
It is calculated by multiplying the z-score by the standard error, which is the population standard deviation divided by the square root of the sample size.
In our scenario, the population standard deviation is 0.62, and with a sample size of 36, the standard error is quite small thanks to the larger sample size. The z-score's multiplication provides us the margin of error, which in this case was approximately 0.266.
This value tells us that the true average GPA is within 0.266 points of our sample mean with 99% confidence.
Z-Score
The z-score is used to derive the margin of error and is critical for constructing the confidence interval. Z-scores relate to how many standard deviations a data point is from the mean in a standard normal distribution, functioning to standardize scores on different scales.
For a 99% confidence interval, the associated z-score, which can be looked up in a z-table, is approximately 2.576.
This number is key since it tells us how many standard deviations our mean falls within the true population mean with a given level of confidence.
Using the z-score ensures that our confidence interval accurately reflects the reliability of our estimate.
Population Standard Deviation
The population standard deviation helps assess the dispersion of a set of data from the mean. It indicates how much variation or "spread" there is from the average. When we assume a known standard deviation, it provides a foundation for estimating the population mean through the sample data.
In the GPA exercise, the population standard deviation was given as 0.62. This value represents the typical deviation of individual freshmen GPAs from the true mean GPA.
Utilizing this known value allows us to calculate the standard error and then proceed with determining the margin of error and eventually the confidence interval.
The population standard deviation is crucial for these steps since it influences how confident we can be that our sample mean approximates the true population mean.

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