Skip to main content
Ch. 5 - Normal Probability Distributions
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 5 - Normal Probability DistributionsProblem 5.R.57a
Chapter 5, Problem 5.R.57a

In Exercises 55–60, find the indicated probabilities and interpret the results.


The mean ACT composite score in a recent year is 20.7. A random sample of 36 ACT composite scores is selected. What is the probability that the mean score for the sample is (a) less than 22, (b) greater than 23, and (c) between 20 and 21.5? Assume sigma=5.9.

Verified step by step guidance
1
Step 1: Identify the given information. The population mean (μ) is 20.7, the population standard deviation (σ) is 5.9, and the sample size (n) is 36. Since the sample size is large (n ≥ 30), the sampling distribution of the sample mean can be approximated by a normal distribution with mean μ and standard error (SE) calculated as SE = σ / √n.
Step 2: Calculate the standard error (SE) of the sampling distribution. Use the formula SE = σ / √n, where σ = 5.9 and n = 36. This will give you the standard deviation of the sampling distribution of the sample mean.
Step 3: Standardize the values for each part of the problem using the z-score formula. The z-score formula is z = (X̄ - μ) / SE, where X̄ is the sample mean, μ is the population mean, and SE is the standard error. For part (a), calculate the z-score for X̄ = 22. For part (b), calculate the z-score for X̄ = 23. For part (c), calculate the z-scores for X̄ = 20 and X̄ = 21.5.
Step 4: Use the standard normal distribution table (or a statistical software) to find the probabilities corresponding to the z-scores calculated in Step 3. For part (a), find the probability that the z-score is less than the value calculated for X̄ = 22. For part (b), find the probability that the z-score is greater than the value calculated for X̄ = 23. For part (c), find the probability that the z-score is between the values calculated for X̄ = 20 and X̄ = 21.5.
Step 5: Interpret the results. For each part, explain the probability in the context of the problem. For example, for part (a), the probability represents the likelihood that the mean score of a random sample of 36 ACT composite scores is less than 22. Similarly, interpret the results for parts (b) and (c).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Central Limit Theorem

The Central Limit Theorem states that the distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the original population distribution. This theorem is crucial for calculating probabilities related to sample means, especially when the sample size is sufficiently large, such as the sample of 36 ACT scores in this question.
Recommended video:
Guided course
04:52
Calculating the Mean

Standard Error of the Mean

The Standard Error of the Mean (SEM) quantifies how much the sample mean is expected to vary from the true population mean. It is calculated by dividing the population standard deviation (sigma) by the square root of the sample size. In this case, with sigma = 5.9 and a sample size of 36, the SEM will help determine the probability of the sample mean falling within specified ranges.
Recommended video:
Guided course
04:52
Calculating the Mean

Z-scores

A Z-score measures how many standard deviations an element is from the mean of a distribution. It is calculated by subtracting the mean from the sample mean and then dividing by the standard error. Z-scores are essential for finding probabilities in a normal distribution, allowing us to determine the likelihood of the sample mean being less than, greater than, or between certain values.
Recommended video:
Guided course
06:31
Z-Scores From Given Probability - TI-84 (CE) Calculator
Related Practice
Textbook Question

In Exercises 69 and 70, determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities.


A survey of U.S. adults found that 72% used a mobile device to manage their bank account at least once in the previous month. You randomly select 70 U.S. adults and ask whether they used a mobile device to manage their bank account at least once in the previous month. Find the probability that the number who have done so is (c) greater than 60.

Textbook Question

In Exercises 55–60, find the indicated probabilities and interpret the results.


Refer to Exercise 34. A random sample of six days is selected. Find the probability that the mean surface concentration of carbonyl sulfide for the sample is (c) more than 11.1 picomoles per liter. Compare your answers with those in Exercise 34.

Textbook Question

In Exercises 53 and 54, find the mean and standard deviation of the indicated sampling distribution of sample means. Then sketch a graph of the sampling distribution.


The test scores for the Law School Admission Test (LSAT) in a recent year are normally distributed, with a mean of 151.88 and a standard deviation of 9.95. Random samples of size 40 are drawn from this population, and the mean of each sample is determined.

Textbook Question

In Exercises 51 and 52, a population and sample size are given. (a) Find the mean and standard deviation of the population.

The goals scored in a season by the four starting defenders on a soccer team are 1, 2, 0, and 3. Use a sample size of 2.

Textbook Question

In Exercises 55–60, find the indicated probabilities and interpret the results.


The mean annual salary for Level 1 actuaries in the United States is about \$72,000. A random sample of 45 Level 1 actuaries is selected. What is the probability that the mean annual salary of the sample is (b) more than \(68,000? Assume sigma = \)11,000.

Textbook Question

In Exercises 69 and 70, determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities.


A survey of U.S. adults found that 72% used a mobile device to manage their bank account at least once in the previous month. You randomly select 70 U.S. adults and ask whether they used a mobile device to manage their bank account at least once in the previous month. Find the probability that the number who have done so is (b) exactly 50.