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.1c

Chapter 5, Problem 5.T.1c

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%?

Verified step by step guidance

Step 1: Identify the key parameters of the problem. The population mean (μ) is 7.46%, the population standard deviation (σ) is 53.47%, and the sample size (n) is 50. The goal is to find the probability that the sample mean (x̄) is between -10% and 30%.

Step 2: Calculate the standard error of the mean (SE). The formula for the standard error is:

SE = \frac{σ}{\sqrt{n}}

. Substitute the values of σ = 53.47% and n = 50 into the formula.

Step 3: Standardize the bounds of the interval (-10% and 30%) to z-scores using the formula:

z = \frac{x̄ - μ}{SE}

. Compute the z-scores for both -10% and 30% by substituting the respective values of x̄, μ, and SE.

Step 4: Use the standard normal distribution table (or a statistical software) to find the cumulative probabilities corresponding to the z-scores calculated in Step 3. These cumulative probabilities represent the area under the standard normal curve up to each z-score.

Step 5: Subtract the cumulative probability of the lower z-score (corresponding to -10%) from the cumulative probability of the upper z-score (corresponding to 30%). This difference gives the probability that the sample mean is between -10% and 30%.

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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 population's distribution. This is crucial for calculating probabilities related to sample means, especially when dealing with larger samples, such as the size of 50 in this question.

Standard Error

Standard Error (SE) measures the dispersion of sample means around the population mean. It is calculated by dividing the population standard deviation by the square root of the sample size. In this case, the SE will help determine the probability of the sample mean falling within a specific range, such as between -10% and 30%.

Z-scores

A Z-score indicates how many standard deviations an element is from the mean. It is calculated by subtracting the mean from the sample mean and 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 within the specified range.

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Related Practice

Textbook Question

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%?

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

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 (a) exactly 7. Identify any unusual events. Explain.

Textbook Question

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 (b) less than 5. Identify any unusual events. Explain.

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?