Question

the salaries of pharmacy techs are normally distributed with a mean of $33,000 and a standard deviation of $4,000. if 30 pharmacy techs are selected at random, what is the probability their average salary is less than $34,200?

Answer 1

To find the probability that the average salary of 30 randomly selected pharmacy techs is less than $34,200, calculate the z-score using the formula z = (x - μ) / (σ / √n), then use the standard normal distribution table to find the corresponding probability. In this case, the probability is approximately 0.9554 or 95.54%.

Step-by-step explanation:

To find the probability that the average salary of 30 randomly selected pharmacy techs is less than $34,200, we need to calculate the z-score and use the standard normal distribution table.

The formula to calculate the z-score is:

z = (x - μ) / (σ / √n)

Where x is the desired average salary, μ is the mean, σ is the standard deviation, and n is the number of pharmacy techs selected.

In this case, x = $34,200, μ = $33,000, σ = $4,000, and n = 30.

Plugging in the values, we get:

z = ($34,200 - $33,000) / ($4,000 / √30)

Simplifying, we find:

z ≈ 1.69

Using the standard normal distribution table, we can find that the probability of z < 1.69 is approximately 0.9554.

So the probability that the average salary of 30 randomly selected pharmacy techs is less than $34,200 is approximately 0.9554 or 95.54%.

Answer 2

The probability that the average salary of 30 randomly selected pharmacy techs is less than $34,200 is 4.95%, calculated using the standard error and finding the z-score corresponding to the sample mean.

Step-by-step explanation:

To calculate the probability that the average salary of 30 randomly selected pharmacy techs is less than $34,200, we need to use the sampling distribution of the sample mean. Since the salaries are normally distributed, and we're dealing with a sample size of 30, which is greater than 30, we can assume that the sampling distribution of the sample mean is also normally distributed (Central Limit Theorem). The mean of the sampling distribution is the same as the population mean, which is $33,000, and the standard deviation (often called the standard error) of the sampling distribution is the population standard deviation divided by the square root of the sample size (n).

The standard error (SE) is calculated as follows:

SE = $4,000 / √30 = $4,000 / 5.477 = approximately $730.30

We then calculate the z-score that corresponds to a sample mean of $34,200:

z = ($34,200 - $33,000) / $730.30 ≈ 1.65

Now we look up the z-score in a standard normal distribution table (or use a calculator) to find the probability of having a z-value less than 1.65, which gives us approximately 0.9505. However, since we are looking for the probability of the average being less than $34,200, we want the probability to the left of the z-value, so we use 1 - 0.9505 = 0.0495, or 4.95%.

Therefore, there is a 4.95% probability that the average salary of 30 randomly selected pharmacy techs is less than $34,200.