The Economic Policy Institute periodically issues reports on worker’s wages. The institute reported that mean wages for male college graduates were $37.39 per hour and for female college graduates were $27.83 per hour in 2017. Assume the standard deviation for male graduates is $4.60, and for female graduates it is $4.10. a. What is the probability that a sample of 50 male graduates will provide a sample mean within $1.00 of the population mean, $37.39?
b. What is the probability that a sample of 50 female graduates will provide a sample mean within $1.00 of the population mean, $27.83?
c. In which of the preceding two cases, part (a) or part (b), do we have a higher probability of obtaining a sample estimate within $1.00 of the population mean? Why?
Z = (score-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability for each.
As for why, look at SEm for each.
a. Well, if we want to calculate the probability that a sample of 50 male graduates will provide a sample mean within $1.00 of the population mean, it means we need to find the probability that the sample mean falls within the range of $36.39 to $38.39.
To do this, we can use the standard deviation of the population, which is $4.60, and the sample size, which is 50. The formula you can use is:
P(mean - margin of error < sample mean < mean + margin of error) = P(sample mean - margin of error < mean < sample mean + margin of error)
Plugging in the values, we get:
P(37.39 - 1.00 < sample mean < 37.39 + 1.00) = P(36.39 < sample mean < 38.39)
Now, we need to calculate the z-scores for the lower and upper bounds:
z_lower = (36.39 - 37.39) / (4.60 / sqrt(50))
z_upper = (38.39 - 37.39) / (4.60 / sqrt(50))
Using a z-table, we can find the probabilities associated with these z-scores. Subtracting the probability corresponding to the lower bound from the probability corresponding to the upper bound will give us the probability we're looking for.
b. Similarly, if we want to calculate the probability that a sample of 50 female graduates will provide a sample mean within $1.00 of the population mean, we need to find the probability that the sample mean falls within the range of $26.83 to $28.83.
Using the standard deviation of the population, $4.10, and the sample size, 50, we can calculate the margins of error, z-scores, and probabilities just like we did for the male graduates.
c. Now, which of these two scenarios has a higher probability? Well, to determine that, we need to compare the probabilities we calculated for part (a) and part (b).
Whichever one has a higher probability is the one where we have a higher chance of obtaining a sample estimate within $1.00 of the population mean.
So, let's crunch the numbers and find out!
To solve this problem, we will use the t-distribution since the population standard deviation is unknown. Here's how we can calculate the probabilities:
a. For a sample of 50 male graduates, we need to find the probability that the sample mean is within $1.00 of the population mean ($37.39). First, we need to calculate the standard error of the mean (SEM) using the formula:
SEM = standard deviation / sqrt(sample size)
For male graduates:
SEM = $4.60 / sqrt(50)
Next, we need to find the t-score for a sample mean that is within $1.00 of the population mean. We can calculate this using the formula:
t-score = (sample mean - population mean) / SEM
For male graduates:
t-score = ($37.39 - $37.39) / SEM
Since the t-score is 0, the probability of obtaining a sample mean within $1.00 of the population mean is equal to the probability that the t-distribution value lies between -1.00 and 1.00. We can use a t-distribution table or a statistical software to find this probability.
b. The process is the same for female graduates. For a sample of 50 female graduates, the probability of obtaining a sample mean within $1.00 of the population mean ($27.83) can be calculated using the same formula as above.
c. To determine which case gives a higher probability, we need to compare the probabilities calculated in parts (a) and (b). The case with the larger probability is the one where we have a higher chance of obtaining a sample estimate within $1.00 of the population mean.
Note: Since the calculations involve finding t-distribution probabilities, it is not possible to provide the exact probabilities without specific t-distribution values or a statistical software.
To solve this problem, we need to use the concept of sampling distributions and the standard error of the mean. Let's break down each part of the question:
a. What is the probability that a sample of 50 male graduates will provide a sample mean within $1.00 of the population mean, $37.39?
To calculate this probability, we need to find the standard error of the mean (SEM) for the sample. The SEM is calculated as the standard deviation divided by the square root of the sample size:
SEM = standard deviation / √sample size
For the male graduates:
SEM = $4.60 / √50 ≈ $0.649
Now, to find the probability that the sample mean will be within $1.00 of the population mean, we need to calculate the z-score. The z-score is the number of standard deviations away from the population mean:
z = (sample mean - population mean) / SEM
In this case, we want the sample mean to be within $1.00 of the population mean, so we have an interval of $37.39 ± $1.00. The z-score for this interval can be calculated as:
z = ($37.39 - $37.39) / $0.649
Since the numerator is zero, the z-score will be zero. The probability of obtaining a sample mean within this interval can be found using a standard normal distribution table or calculator. For a z-score of zero, the probability is 0.5. Therefore, the probability that a sample of 50 male graduates will provide a sample mean within $1.00 of the population mean is 0.5.
b. What is the probability that a sample of 50 female graduates will provide a sample mean within $1.00 of the population mean, $27.83?
Using the same approach as in part (a), we need to find the SEM for the female graduates:
SEM = $4.10 / √50 ≈ $0.579
Calculating the z-score for a sample mean within $1.00 of the population mean:
z = ($27.83 - $27.83) / $0.579 = 0
Again, for a z-score of zero, the probability is 0.5. Therefore, the probability that a sample of 50 female graduates will provide a sample mean within $1.00 of the population mean is also 0.5.
c. In which of the preceding two cases, part (a) or part (b), do we have a higher probability of obtaining a sample estimate within $1.00 of the population mean? Why?
In both cases, the probability is the same, with both being 0.5. This means that the chances of obtaining a sample estimate within $1.00 of the population mean are equal for both male and female graduates.
Therefore, in terms of the probability of obtaining a sample estimate within $1.00 of the population mean, there is no difference between male and female graduates.