Question

1. in 2019, the percent of total labor force of the world that were women was about 39%. the united nations wonders if the percentage of females in the labor force is the same in the united states. they select a random sample of 525 employment records and find that 229 of the people are female.

a. create a 90% confidence interval. write all calculator commands - no hand calculations allowed. (3 points)
b. interpret the confidence interval in this context. (2 points)
c. use your interval to find the margin of error associated with this confidence interval. (1 point)
d. should the representatives from the department of labor conclude that the percentage of females in their labor force is higher than the worlds rate of 39%? explain. (2 points)

1. a survey of shoppers is planned to see what percentage use credit cards. prior surveys suggest that 54% of shoppers use credit cards. how many randomly selected shoppers must we survey to estimate the proportion of shoppers who use credit cards to within 3% with 98% confidence? be sure to show all your work and round appropriately. (4 points)

Explanation:

a. Step - by - Step Format

Step1: Calculate sample proportion

The sample proportion $\hat{p}=\frac{x}{n}$, where $x = 229$ and $n=525$. So, $\hat{p}=\frac{229}{525}\approx0.436$.

Step2: Calculator commands

On a TI - 84 Plus calculator:

• Press STAT
• Arrow over to TESTS
• Arrow down to 1 - PropZInt
• Enter x = 229, n = 525, C - Level=0.90
• Press Calculate

The confidence interval is approximately $(0.400, 0.472)$

b. Answer - Explanation Format

A 90% confidence interval means that if we were to take many random samples of size 525 from the labor force in the United States, about 90% of those samples would produce a confidence interval that contains the true proportion of females in the labor force. In this context, we are 90% confident that the true proportion of females in the U.S. labor force lies between 0.400 and 0.472.

The world rate is 0.39. Since the entire 90% confidence interval $(0.400, 0.472)$ is above 0.39, we can conclude that the proportion of females in the U.S. labor force is higher than the world rate of 39% at the 90% confidence level.

Answer:

We are 90% confident that the true proportion of females in the U.S. labor force lies between 0.400 and 0.472.

c. Step - by - Step Format

Step1: Formula for margin of error

The margin of error $E=\frac{upper - bound - lower - bound}{2}$

Step2: Calculate margin of error

Given the interval $(0.400, 0.472)$, $E=\frac{0.472 - 0.400}{2}=0.036$

d. Answer - Explanation Format