Question

The article in Time magazine \(^{7}\) (Exercise 6.52 ) also reported that \(80 \%\) of men and \(62 \%\) of women put in more than 40 hours a week on the job. Assume that these percentages are correct for all Americans, and that a random sample of 50 working women is selected. a. What is the average number of women who put in more than 40 hours a week on the job? b. What is the standard deviation for the number of women who put in more than 40 hours a week on the job? c. Suppose that in our sample of 50 working women, there are 25 who work more than 40 hours a week. Would you consider this to be an unusual occurrence? Explain.

Short answer

Answer: No, having 25 out of 50 women working more than 40 hours a week is not considered an unusual occurrence.

Step-by-step solution

Use the given percentage

According to the problem, 62% of women work more than 40 hours a week. This means 0.62 of the entire population of women works more than 40 hours a week.

Calculate the average number of women in the sample

In order to find the average number of women who work more than 40 hours a week, we need to multiply the total number of women in the sample (50) by 0.62 (the percentage of women who work more than 40 hours a week). The average can be calculated as follows: Average = (0.62) * (50) = 31 #a. Answer:# The average number of women who put in more than 40 hours a week on the job is 31. #b. Finding the standard deviation for the number of women who work more than 40 hours a week#

Use the binomial distribution formula

For this problem, we can use the binomial distribution formula to find the standard deviation. The standard deviation (σ) can be found using the formula: σ = sqrt(n * p * (1 - p)) Here, 'n' refers to the sample size (50 women), 'p' represents the probability of success (in this case, women working more than 40 hours a week, which is 0.62), and (1 - p) is the probability of failure (in this case, women not working more than 40 hours a week, which is 0.38).

Calculate the standard deviation

Using the formula, we can calculate the standard deviation as follows: σ = sqrt(50 * 0.62 * 0.38) = sqrt(11.740) ≈ 3.426 #b. Answer:# The standard deviation for the number of women who put in more than 40 hours a week on the job is approximately 3.426. #c. Determining if having 25 out of 50 women working more than 40 hours a week is an unusual occurrence#

Calculate the z-score

To determine whether having 25 out of 50 women working more than 40 hours a week is unusual, we need to find the z-score. The z-score can be found using the formula: z = (x - μ) / σ Here, 'x' refers to the observed value (25), 'μ' represents the mean (31), and 'σ' is the standard deviation (3.426).

Find the z-score

Using the formula, we can calculate the z-score as follows: z = (25 - 31) / 3.426 ≈ -1.75

Determine if the z-score indicates an unusual occurrence

Generally, a z-score between -2 and 2 is considered to be normal (not unusual). In this case, the z-score is -1.75, which falls within this range. Therefore, having 25 out of 50 women working more than 40 hours a week is not an unusual occurrence. #c. Answer:# No, having 25 out of 50 women working more than 40 hours a week is not considered an unusual occurrence.

Key concepts

Understanding Probability and Statistics

Probability and statistics play a fundamental role in interpreting data and making decisions based on that data. The study of statistics allows us to collect, analyze, interpret, present, and organize data in a meaningful way. Probability, a key concept within this field, measures the likelihood of a specific event occurring.

In the context of our exercise, probability was used to predict the number of women working more than 40 hours a week based on a given percentage. This exemplifies statistical analysis where, through probability, we can make assertions about populations based on sample data.

Average Calculation in Context

The average, or mean, is a fundamental concept in statistics representing the central or typical value in a set of data. It is obtained by summing all the numbers in a data set and dividing by the count of those numbers.

In our exercise, the average was calculated for a sample of women working over 40 hours a week. The process involved multiplying the percentage of women known to work over 40 hours by the sample size, simplifying complex data into an understandable figure that describes a normal expectation within the group.

Determining Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.

During our statistical exercise, we computed the standard deviation to assess the variability within the number of women who work more than 40 hours per week. By applying the binomial distribution formula, we quantify the expected deviation from the average, providing us with insights on consistency and predictability in work habits.

Interpreting the Z-Score

The z-score is a statistical metric that describes a value's relationship to the mean of a group of values, expressed in terms of standard deviations. It helps us understand whether an observation is typical or atypical within a given data set.

In the exercise, we used the z-score to determine if having 25 out of 50 women working more than 40 hours a week is unusual. By seeing that the z-score falls within the range considered 'normal', we reinforce the notion that for this population and sample size, such a finding is not extraordinary, thus providing reassurance about the reliability of the distribution's prediction.