A leading magazine (like Barron's) reported at one time that the mean number of weeks an individual is unemployed is 15.3 weeks. Assume that the population of all unemployed individuals is normally distributed with the population mean length of unemployment is 15.3 weeks and that the population standard deviation is 9.4 weeks. Suppose you would like to select a random sample of 171 unemployed individuals for a follow-up study.
Enter your answers as numbers accurate to 4 decimal places.
Find the probability that a single randomly selected value is between 15.7 and 16.5.
Answer = $\square$
Find the probability that a sample of size [tex]$n=171$[/tex] is randomly selected with a mean between 15.7 and 16.5.
Answer = $\square$
Answer (2)
The probability that a single randomly selected value is between 15.7 and 16.5 weeks is approximately 0.0339. The probability that the mean of a sample of size 171 is between these values is approximately 0.2401.
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Calculate z-scores for single values 15.7 and 16.5: z 1 ≈ 0.0426 , z 2 ≈ 0.1277 .
Find the probability for a single value: P ( 15.7 < X < 16.5 ) ≈ 0.0338 .
Calculate z-scores for sample means 15.7 and 16.5: z 1 ≈ 0.5572 , z 2 ≈ 1.6715 .
Find the probability for the sample mean: P ( 15.7 < X ˉ < 16.5 ) ≈ 0.2415 .
Explanation
Understand the problem and provided data We are given that the population of unemployed individuals is normally distributed with a mean of μ = 15.3 weeks and a standard deviation of σ = 9.4 weeks. We want to find the probability that a single randomly selected value is between 15.7 and 16.5 weeks, and the probability that the mean of a sample of size n = 171 is between 15.7 and 16.5 weeks.
Calculate the probability for a single value First, let's find the probability that a single randomly selected value X is between 15.7 and 16.5. We need to calculate P ( 15.7 < X < 16.5 ) . To do this, we'll convert these values to z-scores using the formula z = σ x − μ .
For x 1 = 15.7 , the z-score is: z 1 = 9.4 15.7 − 15.3 = 9.4 0.4 ≈ 0.0426
For x 2 = 16.5 , the z-score is: z 2 = 9.4 16.5 − 15.3 = 9.4 1.2 ≈ 0.1277
Now we need to find the area under the standard normal curve between these two z-scores. Using a standard normal distribution table or a calculator, we find:
P ( z < 0.0426 ) ≈ 0.5170 P ( z < 0.1277 ) ≈ 0.5508
So, the probability that a single randomly selected value is between 15.7 and 16.5 is:
P ( 15.7 < X < 16.5 ) = P ( 0.0426 < z < 0.1277 ) = 0.5508 − 0.5170 = 0.0338
Calculate the probability for the sample mean Next, let's find the probability that the mean of a sample of size n = 171 is between 15.7 and 16.5. The standard deviation of the sample mean is given by σ X ˉ = n σ = 171 9.4 ≈ 0.7179 .
We need to calculate P ( 15.7 < X ˉ < 16.5 ) . To do this, we'll convert these values to z-scores using the formula z = σ X ˉ x ˉ − μ .
For x ˉ 1 = 15.7 , the z-score is: z 1 = 0.7179 15.7 − 15.3 = 0.7179 0.4 ≈ 0.5572
For x ˉ 2 = 16.5 , the z-score is: z 2 = 0.7179 16.5 − 15.3 = 0.7179 1.2 ≈ 1.6715
Now we need to find the area under the standard normal curve between these two z-scores. Using a standard normal distribution table or a calculator, we find:
P ( z < 0.5572 ) ≈ 0.7112 P ( z < 1.6715 ) ≈ 0.9527
So, the probability that the sample mean is between 15.7 and 16.5 is:
P ( 15.7 < X ˉ < 16.5 ) = P ( 0.5572 < z < 1.6715 ) = 0.9527 − 0.7112 = 0.2415
State the final answer Therefore, the probability that a single randomly selected value is between 15.7 and 16.5 is approximately 0.0338, and the probability that a sample of size n = 171 is randomly selected with a mean between 15.7 and 16.5 is approximately 0.2415.
Examples
Understanding probabilities related to unemployment rates can help economists and policymakers assess the effectiveness of employment programs. For instance, if a new job training program is implemented, analyzing the probability of individuals being unemployed for a certain period can provide insights into the program's impact. By comparing the unemployment durations before and after the program, policymakers can determine whether the program is successfully reducing unemployment and adjust strategies accordingly. This type of analysis can also help in forecasting future unemployment trends and informing decisions about resource allocation and economic planning.
