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Chapter 19: Problem 18

Suppose the population of grade-point averages (GPAs) for students at the end of their first year at a large university has a mean of 3.1 and a standard deviation of \(.5 .\) Draw a picture of the frequency curve for the mean GPA of a random sample of 100 students, similar to Figure \(19.5 .\)

Short Answer

Expert verified
The frequency curve is a normal distribution centered around 3.1 with a standard deviation of 0.05.

Step by step solution

01

Understand the Problem

We are tasked with drawing a frequency curve for the mean GPA of a random sample of 100 students. We know the population mean is 3.1 and the population standard deviation is 0.5.
02

Identify the Type of Distribution

Since we are considering the mean GPA of a sample, we will use the Central Limit Theorem. For a sufficiently large sample size (n=100), the sampling distribution of the sample mean will be approximately normal.
03

Determine the Sampling Distribution Parameters

The mean of the sampling distribution of the sample mean is equal to the population mean, which is 3.1. The standard deviation of the sampling distribution, also called the standard error, is calculated using the formula \( \sigma_\bar{x} = \frac{\sigma}{\sqrt{n}} \), where \( \sigma \) is the population standard deviation and \( n \) is the sample size. This becomes \( \frac{0.5}{\sqrt{100}} = 0.05 \).
04

Sketch the Frequency Curve

Draw a normal distribution curve centered at the sample mean (3.1) with a standard deviation of 0.05. The x-axis represents the mean GPAs, with the center of the curve at 3.1. Below the x-axis, scale from slightly below 3.1 - (3*0.05) = 2.95 to slightly above 3.1 + (3*0.05) = 3.25, encompassing about 99.7% of the data as predicted by the Empirical Rule.

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Key Concepts

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

Sampling Distribution
Sampling distribution is a crucial concept in statistics. It refers to the probability distribution of a specific statistic, such as the mean, obtained from a large number of samples drawn from a specific population.
For example, if you repeatedly take samples of 100 students' GPAs from the large university, and each time calculate the average GPA of the sample, the frequency with which each average GPA occurs forms the sampling distribution.

Key characteristics of sampling distribution include:
  • It shows how the sample statistics (like the mean) behave as you take different samples.
  • With a large enough sample size, the sampling distribution will be centered around the true population parameter (e.g., the mean). In our case, that's 3.1.
  • It generally forms a bell-shaped curve (normal distribution), especially when the sample size is large due to the Central Limit Theorem.
This distribution helps statisticians make inferences about the population based on sample data.
Standard Error
The standard error (SE) tells us how much the sample statistic, like the mean of a sample, is expected to vary from the actual population mean. It is essentially the standard deviation of the sampling distribution.
The formula for the standard error of the sample mean is given by:

\[ \sigma_\bar{x} = \frac{\sigma}{\sqrt{n}} \]

Here,
  • \( \sigma \) is the population standard deviation.
  • \( n \) is the sample size.
So for our GPA example, the standard error is \( \frac{0.5}{\sqrt{100}} = 0.05 \).

This means that the average GPA from a sample of 100 students is expected to deviate from the population mean (3.1) by about 0.05. Standard error provides us with an idea of the precision of the sample mean estimate.
Normal Distribution
A normal distribution, often referred to as a bell curve, is a common pattern for distribution of data.
It's symmetric around the mean, where the left side is a mirror image of the right side, and most occurrences take place close to the mean.

Key features include:
  • Mean, median, and mode are all equal and located at the center of the distribution.
  • The distribution is fully defined by its mean and standard deviation.
  • About 68% of data falls within one standard deviation from the mean, 95% falls within two, and 99.7% falls within three, according to the Empirical Rule.
In our GPA example, since we're dealing with a sample mean of 100 students, thanks to the Central Limit Theorem, the distribution of the sample mean approximates a normal distribution centered at the mean GPA, which is 3.1, with a standard deviation (standard error) of 0.05. This makes the normal curve an ideal tool to visualize and make statistical inferences in this scenario.

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Most popular questions from this chapter

The administration of a large university wants to use a random sample of students to measure student opinion of a new food service on campus. Administrators plan to use a continuous scale from 1 to \(100,\) where 1 is complete dissatisfaction and 100 is complete satisfaction. They know from past experience with such questions that the standard deviation for the responses is going to be about \(5,\) but they do not know what to expect for the mean. They want to be almost sure that the sample mean is within plus or minus 1 point of the true population mean value. How large will their random sample have to be?

Use the Rule for Sample Means to explain why it is desirable to take as large a sample as possible when trying to estimate a population value.

According to the Sacramento Bee (2 April 1998, p. F5), Americans get an average of 6 hours and 57 minutes of sleep per night. A survey of a class of 190 statistics students at a large university found that they averaged 7.1 hours of sleep the previous night, with a standard deviation of 1.95 hours. a. Assume that the population average for adults is 6 hours and 57 minutes, or 6.95 hours of sleep per night, with a standard deviation of 2 hours. Draw a picture similar to Figure 19.5 illustrating how the Rule for Sample Means would apply to sample means for random samples of 190 adults. b. Would the mean of 7.1 hours of sleep obtained from the statistics students be a reasonable value to expect for the sample mean of a random sample of 190 adults? Explain. c. Can the sample taken in the statistics class be considered to be a representative sample of all adults? Explain.

Suppose \(20 \%\) of all television viewers in the country watch a particular program. a. For a random sample of 2500 households measured by a rating agency, describe the frequency curve for the possible sample proportions who watch the program. b. The program will be canceled if the ratings show less than \(17 \%\) watching in a random sample of households. Given that 2500 households are used for the ratings, is the program in danger of getting canceled? Explain. c. Draw a picture of the possible sample proportions, similar to Figure \(19.3 .\) Illustrate where the sample proportion of .17 falls on the picture. Use this to confirm your answer in part (b).

Suppose the population of IQ scores in the town or city where you live is bell-shaped, with a mean of 105 and a standard deviation of \(15 .\) Describe the frequency curve for possible sample means that would result from random samples of 100 IQ scores.
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