a. Calculate the standard error of the mean.
b. What is the probability that the sample mean will be less than 84?
c. What is the probability that the sample mean will be more than 83?
d. What is the probability that the sample mean will be between 81.5 and 83.5?
a. The standard error of the mean is (Round to two decimal places as needed.)
The correct answer and explanation is :
To answer this question properly, I need the data or parameters that we can use to calculate the standard error of the mean and probabilities. Typically, these parameters would include:
- The population standard deviation (σ)
- The sample size (n)
- The population mean (μ)
If you have these values, I can calculate the standard error of the mean (SE), and based on the standard error, I can also determine the probabilities associated with the sample mean for the given distribution.
Let’s break down the calculations:
a. Standard Error of the Mean (SE)
The formula for the standard error of the mean (SE) is:
[
SE = \frac{\sigma}{\sqrt{n}}
]
Where:
- ( \sigma ) is the population standard deviation.
- ( n ) is the sample size.
Once the standard error is computed, we can then move on to the next parts of the problem.
b. Probability the sample mean is less than 84
To find the probability that the sample mean will be less than 84, we typically use the Z-score formula:
[
Z = \frac{X – \mu}{SE}
]
Where:
- ( X ) is the value we are interested in (84 in this case).
- ( \mu ) is the population mean.
- ( SE ) is the standard error.
Once we have the Z-score, we can use standard Z-tables or normal distribution calculators to determine the probability.
c. Probability the sample mean is more than 83
This can be calculated in a similar way, but this time for the probability of getting a value greater than 83. Again, we compute the Z-score and then find the corresponding probability from a Z-table.
d. Probability the sample mean is between 81.5 and 83.5
We compute the Z-scores for both values (81.5 and 83.5), and then find the area between them on the normal distribution curve. This gives the probability of the sample mean lying between these two values.
If you provide the values for population mean (μ), standard deviation (σ), and sample size (n), I can proceed with exact calculations and provide an image of the probability distributions.