Ch. 5 - Normal Probability Distributions

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 5 - Normal Probability Distributions

Problem 5.RS.2a

Chapter 5, Problem 5.RS.2a

Assume the machine shifts and is filling the vials with a mean amount of 9.96 milligrams and a standard deviation of 0.05 milligram. You select five vials and find the mean amount of compound added.

a. What is the probability that you select a sample of five vials that has a mean that is within the acceptable range? (See figure.)

Verified step by step guidance

Step 1: Identify the key parameters of the problem. The mean of the shifted machine distribution is μ = 9.96 milligrams, the standard deviation is σ = 0.05 milligrams, and the sample size is n = 5. The acceptable range is visually indicated in the figure.

Step 2: Calculate the standard error of the mean (SEM) for the sample of size n = 5 using the formula SEM = σ / √n. This will give the variability of the sample mean distribution.

Step 3: Determine the z-scores corresponding to the upper and lower limits of the acceptable range. Use the formula z = (X - μ) / SEM, where X is the limit of the acceptable range, μ is the mean, and SEM is the standard error calculated in Step 2.

Step 4: Use the z-scores obtained in Step 3 to find the cumulative probabilities from the standard normal distribution table. These probabilities represent the area under the curve up to the z-scores.

Step 5: Subtract the cumulative probability of the lower limit from the cumulative probability of the upper limit to find the probability that the sample mean falls within the acceptable range.

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Key Concepts

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

Sampling Distribution of the Mean

The sampling distribution of the mean describes the distribution of sample means obtained from a population. When samples of a fixed size are taken, the means of these samples will form their own distribution, which is typically normal if the sample size is sufficiently large, according to the Central Limit Theorem. In this case, the mean of the sampling distribution is equal to the population mean, and its standard deviation is the population standard deviation divided by the square root of the sample size.

Standard Error

Standard error is a measure of the variability of sample means around the population mean. It is calculated as the population standard deviation divided by the square root of the sample size (n). In this scenario, with a standard deviation of 0.05 milligrams and a sample size of 5, the standard error helps determine how much the sample mean is expected to fluctuate, which is crucial for calculating probabilities related to the sample mean.

Probability and Normal Distribution

Probability in statistics often involves determining the likelihood of a certain outcome occurring within a defined range. When dealing with normally distributed data, probabilities can be calculated using z-scores, which standardize the values based on the mean and standard deviation. In this question, the probability of selecting a sample mean within an acceptable range can be found by calculating the area under the normal curve between the specified limits, which is represented in the provided figure.

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Related Practice

Textbook Question

In Exercises 5 and 6, 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. undergraduates found that 37% of those attending in-state colleges would prefer to take a job in a different state after graduation. You randomly select 18 U.S. undergraduates attending in-state colleges. Find the probability that the number who would prefer to take a job in a different state after graduation is (c) at least 10. Identify any unusual events. Explain.

Textbook Question

Assume the machine shifts and is filling the vials with a mean amount of 9.96 milligrams and a standard deviation of 0.05 milligram. You select five vials and find the mean amount of compound added.

c. Which is more sensitive to a shift of parameters—an individual random selection or a randomly selected sample mean?

Textbook Question

Use technology to find the standard deviation of the set of 36 sample means. How does it compare with the standard deviation of the ages found in Exercise 5? Does this agree with the result predicted by the Central Limit Theorem?

Textbook Question

Assume the machine shifts and the distribution of the amount of the compound added now has a mean of 9.96 milligrams and a standard deviation of 0.05 milligram. You select one vial and determine how much of the compound was added.

a. What is the probability that you select a vial that is within the acceptable range (in other words, you do not detect that the machine has shifted)? (See figure.)

Textbook Question

Assume the machine shifts and is filling the vials with a mean amount of 9.96 milligrams and a standard deviation of 0.05 milligram. You select five vials and find the mean amount of compound added.

b. You randomly select three samples of five vials. What is the probability that you select at least one sample of five vials that has a mean that is within the acceptable range?

Textbook Question

In Exercises 2–4, the random variable x is normally distributed with mean mu= 18 and standard deviation sigma 7.6

Find the value of x that has 88.3% of the distribution’s area to its left.