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.T.1b

Chapter 5, Problem 5.T.1b

A certain stock's daily percent return on Fridays has a mean of 3.12% and a standard deviation of 41.25%. If random samples of 40 days are selected and the mean return for each sample is calculated, what is the probability that a sample mean is greater than 17%?

Verified step by step guidance

Step 1: Identify the problem as one involving the sampling distribution of the sample mean. The sample mean follows a normal distribution because the sample size (n = 40) is sufficiently large (Central Limit Theorem applies).

Step 2: Write down the parameters of the population distribution: the population mean (μ = 3.12%) and the population standard deviation (σ = 41.25%).

Step 3: Calculate the standard error of the mean (SE), which is the standard deviation of the sampling distribution of the sample mean. Use the formula:

SE = \frac{σ}{\sqrt{n}}

, where σ is the population standard deviation and n is the sample size.

Step 4: Standardize the sample mean of 17% to a z-score using the formula:

z = \frac{x̄ - μ}{SE}

, where x̄ is the sample mean (17%), μ is the population mean (3.12%), and SE is the standard error calculated in Step 3.

Step 5: Use the z-score obtained in Step 4 to find the probability that the sample mean is greater than 17%. This can be done by looking up the z-score in a standard normal distribution table or using statistical software to find the area to the right of the z-score.

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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 sample means will approach a normal distribution as the sample size increases, regardless of the original distribution's shape. This theorem is crucial for understanding how sample means behave, especially when dealing with larger samples, such as the 40 days in this question.

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 by the square root of the sample size. In this case, it helps determine the variability of the sample means when assessing the probability of exceeding a specific value.

Z-Score

A Z-score measures how many standard deviations an element is from the mean. In this context, it is used to standardize the sample mean to find the probability of it being greater than 17%. By calculating the Z-score, we can utilize the standard normal distribution to determine the likelihood of observing a sample mean above a certain threshold.

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Related Practice

Textbook Question

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

Find each probability.

b. P(0 < x < 5)

Textbook Question

The per capita disposable income for residents of a U.S. city in a recent year is normally distributed, with a mean of about \$44,000 and a standard deviation of about \(2450. Use this information in Exercises 7–10.

Out of 800 residents, about how many would you expect to have a disposable income of between \)40,000 and \$42,000?

Textbook Question

Pregnancy Length Use the normal distribution in Exercise 15.

a. What percent of the new mothers had a pregnancy length of less than 290 days?

Textbook Question

Finding Probabilities for Normal Distributions In Exercises 7–12, find the indicated probabilities. If convenient, use technology to find the probabilities.

MCAT Scores In a recent year, the MCAT total scores were normally distributed, with a mean of 500.9 and a standard deviation of 10.6. Find the probability that a randomly selected medical student who took the MCAT has a total score that is (a) less than 490. Identify any unusual events in parts (a)–(c). Explain your reasoning. (Source: Association of American Medical Colleges)

Textbook Question

During a recent period of one year, the mean percent increase in value on Wednesdays of the cryptocurrency Dogecoin was 7.46%, with a standard deviation of 53.47%. Random samples of size 50 are drawn from this population and the mean of each sample is determined. (Source: Crypto Indicators)

c. What is the probability that the mean percent increase for a given sample is between −10% and 30%?

Textbook Question

Manufacturer Claims You work for a consumer watchdog publication and are testing the advertising claims of a tire manufacturer. The manufacturer claims that the life spans of the tires are normally distributed, with a mean of 40,000 miles and a standard deviation of 4000 miles. You test 16 tires and record the life spans shown below.

a. Draw a frequency histogram to display these data. Use five classes. Do the life spans appear to be normally distributed? Explain.