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Chapter 7: Problem 50

According to data released in 2016 , \(69 \%\) of students in the United States enroll in college directly after high school graduation. Suppose a sample of 200 recent high school graduates is randomly selected. After verifying the conditions for the Central Limit Theorem are met, find the probability that at most \(65 \%\) enrolled in college directly after high school graduation. (Source: nces.ed.gov)

Short Answer

Expert verified
The probability that at most 65% of a randomly selected sample of 200 recent high school graduates enrolled in college directly after graduation is the cumulative probability associated with the z-score calculated in Step 2. This value can be found using a standard normal distribution table or a calculator.

Step by step solution

01

Calculate Mean and Standard Deviation of Sample Proportion

The mean of the sample proportion is equal to the population proportion, $\mu_p = P = 0.69$. The standard deviation of the sample proportion can be calculated using the formula \(\sigma_p = \sqrt{\frac{P(1 - P)}{n}} = \sqrt{\frac{(0.69)(1 - 0.69)}{200}}.\)
02

Standardize Desired Proportion using Z-Score Formula

The z-score for the desired proportion (0.65) can be calculated using the formula \(Z = \frac{p - \mu_p}{\sigma_p}\). Plugging in the known values, \(Z = \frac{0.65 - 0.69}{\sigma_p}\), where \(\sigma_p\) is the value calculated in Step 1.
03

Find Cumulative Probability for Z-Score

The z-score represents how many standard deviations away the desired proportion is from the mean. To find the probability of getting a sample proportion at most 0.65, you need to find the area to the left of this z-score on the standard normal distribution. This can be accomplished using a standard normal (Z) table or a calculator with this functionality.

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Key Concepts

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

Sample Proportion
When analyzing data about a group, the sample proportion is crucial. It refers to the fraction of individuals in a sample with a certain characteristic. In our exercise, we're looking at high school graduates who enrolled in college. From a sample of 200 students, the sample proportion will help us estimate or infer about the larger population.
To calculate it, use the formula:
  • \( \hat{p} = \frac{x}{n} \)
Here, \( x \) is the number of favorable outcomes (students enrolling in college) and \( n \) is the sample size (200 graduates). This estimation is useful for comparing with the overall population proportion.
Z-Score
Standardizing a value in statistics involves calculating the z-score. It's a way of understanding how far a value is from the mean in terms of standard deviations. In our exercise, the z-score helps determine how unusual a sample proportion of 0.65 is compared to the mean 0.69.
To find the z-score, use the formula:
  • \( Z = \frac{p - \mu_p}{\sigma_p} \)
Where \( p \) is the sample proportion (0.65), \( \mu_p \) is the mean sample proportion (0.69), and \( \sigma_p \) is the standard deviation of the sample proportion. Calculating the z-score allows us to translate the problem into the context of a standard normal distribution.
Normal Distribution
The normal distribution is a critical concept in statistics, characterized by its bell-shaped curve. It describes how data is expected to be distributed in many natural phenomena. In our problem, once we standardize our sample proportion using the z-score, we observe its position on the normal distribution.
The normal distribution has properties:
  • Mean, median, and mode are equal.
  • Symmetrical around the mean.
  • Area under the curve represents probability, totaling 1.
The Central Limit Theorem supports using the normal distribution for sample proportions when the sample size is large, as is the case here. This helps us determine the likelihood of observing certain sample proportions.
Probability Calculation
Finding the probability of an event involves calculating the likelihood that a particular outcome will occur. In this context, we're interested in the probability that the sample proportion is at most 0.65.
After computing the z-score, find the cumulative probability. This probability is the area to the left of the z-score on the normal distribution curve. You can use:
  • A standard normal (Z) table, or
  • A statistical calculator
This area represents the probability we seek. It tells us how likely it is for a sample of 200 graduates to have 65% or fewer enroll directly in college.

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Most popular questions from this chapter

The website scholarshipstats.com collected data on all 5341 NCAA basketball players for the 2017 season and found a mean height of 77 inches. Is the number 77 a parameter or a statistic? Also identify the population and explain your choice.

A 2017 survey of U.S. adults found the \(64 \%\) believed that freedom of news organization to criticize political leaders is essential to maintaining a strong democracy. Assume the sample size was 500 . a. How many people in the sample felt this way? b. Is the sample large enough to apply the Central Limit Theorem? Explain. Assume all other conditions are met. c. Find a \(95 \%\) confidence interval for the proportion of U.S. adults who believe that freedom of news organizations to criticize political leaders is essential to maintaining a strong democracy. d. Find the width of the \(95 \%\) confidence interval. Round your answer to the nearest whole percent. e. Now assume the sample size was increased to 4500 and the percentage was still \(64 \%\). Find a \(95 \%\) confidence interval and report the width of the interval. f. What happened to the width of the confidence interval when the sample size was increased. Did it increase or decrease?

a. If a rifleman's gunsight is adjusted incorrectly, he might shoot bullets consistently close to 2 feet left of the bull's-eye target. Draw a sketch of the target with the bullet holes. Does this show lack of precision or bias? b. Draw a second sketch of the target if the shots are both unbiased and precise (have little variation). The rifleman's aim is not perfect, so your sketches should show more than one bullet hole.

A 2016 Pew Research poll found that \(61 \%\) of U.S. adults believe that organic produce is better for health than conventionally grown varieties. Assume the sample size was 1000 and that the conditions for using the CLT are met. a. Find and interpret a \(95 \%\) confidence interval for the proportion of U.S. adults to believe organic produce is better for health. b. Find and interpret an \(80 \%\) confidence interval for this population parameter. c. Which interval is wider? d. What happens to the width of a confidence interval as the confidence level decrease?

Suppose it is known that \(60 \%\) of employees at a company use a Flexible Spending Account (FSA) benefit. a. If a random sample of 200 employees is selected, do we expect that exactly \(60 \%\) of the sample uses an FSA? Why or why not? b. Find the standard error for samples of size 200 drawn from this population. What adjustments could be made to the sampling method to produce a sample proportion that is more precise?
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