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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.56c
Chapter 5, Problem 5.R.56c

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.

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1
Identify the key components of the problem: the sample size (n = 6), the population mean (μ), the population standard deviation (σ), and the threshold value (11.1 picomoles per liter). These values should be referenced from Exercise 34.
Determine the sampling distribution of the sample mean. The mean of the sampling distribution is the same as the population mean (μ), and the standard deviation of the sampling distribution (standard error) is calculated as σ/√n, where n is the sample size.
Standardize the threshold value (11.1) to a z-score using the formula: z = (X̄ - μ) / (σ/√n), where X̄ is the threshold value, μ is the population mean, and σ/√n is the standard error.
Use the standard normal distribution table (or a statistical software) to find the probability corresponding to the calculated z-score. Since the problem asks for the probability that the mean is more than 11.1, calculate the area to the right of the z-score (1 - cumulative probability).
Compare the result with the probabilities calculated in Exercise 34 to interpret how the sample size affects the probability. Larger sample sizes typically result in smaller standard errors, which can influence the probabilities.

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Key Concepts

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

Sampling Distribution

The sampling distribution is the probability distribution of a statistic (like the sample mean) obtained from a large number of samples drawn from a specific population. It describes how the sample mean varies from sample to sample and is crucial for understanding how to calculate probabilities related to sample means, especially when the sample size is small.
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Sampling Distribution of Sample Proportion

Central Limit Theorem (CLT)

The Central Limit Theorem states that, for a sufficiently large sample size, the distribution of the sample mean will approximate a normal distribution, regardless of the population's distribution. This theorem is essential for calculating probabilities related to sample means, particularly when the sample size is small, as it allows us to use normal distribution properties.
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Calculating the Mean

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences or draw conclusions about a population based on sample data. In this context, it involves determining whether the sample mean of carbonyl sulfide concentrations exceeds a specified value (11.1 picomoles per liter) and interpreting the significance of this result in relation to the null hypothesis.
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Step 1: Write Hypotheses
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 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 51 and 52, a population and sample size are given. (b) List all samples (with replacement) of the given size from the population and find the mean of each. (c) Find the mean and standard deviation of the sampling distribution of sample means and compare them with 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 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.

Textbook Question

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.