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Chapter 7: Problem 13

A normal population has a mean of 20.0 and a standard deviation of \(4.0 .\) a. Compute the \(z\) value associated with 25.0 . b. What proportion of the population is between 20.0 and \(25.0 ?\) c. What proportion of the population is less than \(18.0 ?\)

Short Answer

Expert verified
a. The Z-score is 1.25. b. 39.44% of the population is between 20.0 and 25.0. c. 30.85% of the population is less than 18.0.

Step by step solution

01

Calculate the Z-score for 25.0

The Z-score formula is given by \( z = \frac{X - \mu}{\sigma} \), where \( X \) is the value in question, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. Here, \( X = 25.0 \), \( \mu = 20.0 \), and \( \sigma = 4.0 \). Calculate \( z = \frac{25.0 - 20.0}{4.0} = \frac{5.0}{4.0} = 1.25 \).
02

Find the Proportion Between 20.0 and 25.0

To find the proportion of the population between 20.0 and 25.0, we first need the Z-score for both values. The Z-score for 20 is 0 because it is the mean. Using a standard normal distribution table or calculator, find the area to the left of \( z = 1.25 \) (which is approximately 0.8944). Since the mean corresponds to 0.5 (or 50%), subtract: 0.8944 - 0.5 = 0.3944. Therefore, approximately 39.44% of the population is between 20.0 and 25.0.
03

Find the Proportion Less Than 18.0

First, calculate the Z-score for 18.0: \( z = \frac{18.0 - 20.0}{4.0} = \frac{-2.0}{4.0} = -0.50 \). Use a standard normal distribution table to find the area to the left of \( z = -0.50 \), which is approximately 0.3085. Thus, approximately 30.85% of the population is less than 18.0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Z-Score
The z-score is a measure that describes a single data point's position relative to the mean of a data set, expressed in terms of standard deviations. It helps us understand how far from the average a particular value is.

The formula for calculating a z-score is: \[ z = \frac{X - \mu}{\sigma} \]where:
  • \(X\) is the value in question.
  • \(\mu\) is the mean of the distribution.
  • \(\sigma\) is the standard deviation.
If the z-score is positive, the data point is above the mean. If it's negative, the data point is below the mean.

In the exercise, the z-score for 25.0 is calculated as 1.25. This means that 25.0 is 1.25 standard deviations above the average value of 20.0. Z-scores are pivotal for comparing different data sets.
Standard Deviation
Standard deviation is a measure of the amount of variation or dispersion in a set of values. It quantifies how spread out the numbers in your data set are. A small standard deviation means that most of the numbers are close to the mean, while a large standard deviation indicates that the numbers are more spread out.

Standard deviation is often used in statistics to assess the risk and volatility of investments. It tells us how much the data can deviate from the average.
For the exercise given:
  • The mean \(\mu\) is 20.
  • The standard deviation \(\sigma\) is 4.0.
This means, on average, the data points deviate from the mean by 4 units. Understanding standard deviation is crucial for interpreting the variability in a data set and making predictions based on it.
Proportion
The term "proportion" in statistics refers to the fraction of the total population that exhibits a certain characteristic. When calculated using a normal distribution, proportions give us valuable insights into how much of our data falls within a specified range.

Proportion calculations often involve using z-scores and standard normal distribution tables:
  • To find the proportion of the population between 20 and 25, we look between the z-scores of 0 and 1.25. By using a normal distribution table, we find that approximately 39.44% of the population falls within this range.
  • To find the proportion less than 18.0, the z-score is -0.50. The table shows that approximately 30.85% of the population is below this threshold.
Calculating proportions helps us understand how a particular value compares within a dataset, providing deeper insights into the characteristics of that distribution.

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Most popular questions from this chapter

The weights of canned hams processed at Henline Ham Company follow the normal distribution, with a mean of 9.20 pounds and a standard deviation of 0.25 pounds. The label weight is given as 9.00 pounds. a. What proportion of the hams actually weigh less than the amount claimed on the label? b. The owner, Glen Henline, is considering two proposals to reduce the proportion of hams below label weight. He can increase the mean weight to 9.25 and leave the standard deviation the same, or he can leave the mean weight at 9.20 and reduce the standard deviation from 0.25 pounds to \(0.15 .\) Which change would you recommend?

According to the South Dakota Department of Health the mean number of hours of TV viewing per week is higher among adult women than men. A recent study showed women spent an average of 34 hours per week watching TV and men 29 hours per week. Assume that the distribution of hours watched follows the normal distribution for both groups, and that the standard deviation among the women is 4.5 hours and is 5.1 hours for the men. a. What percent of the women watch TV less than 40 hours per week? b. What percent of the men watch TV more than 25 hours per week? c. How many hours of TV do the one percent of women who watch the most TV per week watch? Find the comparable value for the men.

A normal distribution has a mean of 50 and a standard deviation of 4. Determine the value below which 95 percent of the observations will occur.

Most four-year automobile leases allow up to 60,000 miles. If the lessee goes beyond this amount, a penalty of 20 cents per mile is added to the lease cost. Suppose the distribution of miles driven on four-year leases follows the normal distribution. The mean is 52,000 miles and the standard deviation is 5,000 miles. a. What percent of the leases will yield a penalty because of excess mileage? b. If the automobile company wanted to change the terms of the lease so that 25 percent of the leases went over the limit, where should the new upper limit be set? c. One definition of a low-mileage car is one that is 4 years old and has been driven less than 45,000 miles. What percent of the cars returned are considered lowmileaqe?

The net sales and the number of employees for aluminum fabricators with similar characteristics are organized into frequency distributions. Both are normally distributed. For the net sales, the mean is \(\$ 180\) million and the standard deviation is \(\$ 25\) million. For the number of employees, the mean is 1,500 and the standard deviation is \(120 .\) Clarion Fabricators had sales of \(\$ 170\) million and 1,850 employees. a. Convert Clarion's sales and number of employees to \(z\) values. b. Locate the two \(z\) values. c. Compare Clarion's sales and number of employees with those of the other fabricators.
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