To estimate the mean height population of male students on your campus, you will measure and SRS of students. Heights of people the same sex and similiar ages are close to normal. You know from government data that the standard deviation of the heights of young men is about 2.8 inches. Suppose (that unknown to you) the mean heights of all males students is 70 inches.

A. If you choose one student at random, what is the probability that he is between 69 and 78 inches tall?

B. You measure 25 students. What is the sampling distribution of their average height X (sample mean)?

C. What is the probability that the mean height of your sample is between 69 and 71 inches?

I missed my class, and I'm lost on how to do this.

A. Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to your Z -scores.

B. Stand Error of the mean (SEm) = SD/√(n-1)

C. Z = (mean1 - mean2)/standard error (SE) of difference between means

SEdiff = √(SEmean1^2 + SEmean2^2)

SEm = SD/√(n-1)

Since only one SD is provided, you can use just that to determine SEdiff.

Use that same table.

A. To find the probability of a random student being between 69 and 78 inches tall, you will need to convert these values to standard units. Standard units, also known as z-scores, measure how many standard deviations a value is away from the mean.

First, you need to find the z-scores for 69 and 78 inches. The formula for calculating a z-score is:

z = (x - μ) / σ

where
x = value of interest (height in this case)
μ = population mean
σ = population standard deviation

For 69 inches:
z1 = (69 - 70) / 2.8

For 78 inches:
z2 = (78 - 70) / 2.8

Once you have the z-scores, you can use a standard normal distribution table or a calculator to find the probability of a z-score falling between z1 and z2.

B. The sampling distribution of the sample mean (X) can be approximated using a normal distribution if the sample size is large enough (for example, n > 30). In this case, you have measured 25 students, which exceeds the threshold for normal approximation.

The mean of the sampling distribution will be equal to the population mean, which is 70 inches. The standard deviation of the sampling distribution, also known as the standard error, can be calculated using the formula:

SE = σ / sqrt(n)

where
σ = population standard deviation
n = sample size

Since the standard deviation is given as 2.8 inches and the sample size is 25, you can now calculate the standard error (SE).

C. To find the probability of the mean height of your sample falling between 69 and 71 inches, you need to convert these values to standard units as we did in part A.

For 69 inches:
z1 = (69 - 70) / (2.8 / sqrt(25))

For 71 inches:
z2 = (71 - 70) / (2.8 / sqrt(25))

Once you have the z-scores, you can use a standard normal distribution table or a calculator to find the probability of a z-score falling between z1 and z2.

To estimate the mean height population of male students on your campus, you can use statistical methods. Let's break down the questions one by one and explain how to approach each one.

A. If you choose one student at random, what is the probability that he is between 69 and 78 inches tall?
To find this probability, you can use the concept of the standard normal distribution. Since we know that heights are close to normal and the standard deviation is 2.8 inches, we can assume a normal distribution to estimate probabilities.
1. Convert the given heights to z-scores. The formula for calculating the z-score is:
z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.
For example, for 69 inches, the z-score would be (69 - 70) / 2.8.
2. Use a standard normal distribution table or a statistical software to find the probabilities associated with the z-scores.
The probability would be the difference between the cumulative probabilities of the z-scores for 78 inches and 69 inches.
P(69 ≤ X ≤ 78) = P(z1 ≤ Z ≤ z2), where Z represents the z-scores for 69 and 78 inches.

B. You measure 25 students. What is the sampling distribution of their average height X (sample mean)?
The sampling distribution of the sample mean is an important concept in statistics. It represents the distribution of all possible sample means that could be obtained by repeatedly sampling from the same population.
1. The mean of the sampling distribution, denoted as μ(subscript x̄), is equal to the population mean, which in this case is 70 inches.
2. The standard deviation of the sampling distribution, denoted as σ(subscript x̄), is equal to the population standard deviation divided by the square root of the sample size.
σ(subscript x̄) = σ / sqrt(n), where σ is the population standard deviation and n is the sample size.

C. What is the probability that the mean height of your sample is between 69 and 71 inches?
To find this probability, you can use the sampling distribution of the sample mean.
1. Convert the given heights to z-scores using the same formula mentioned earlier.
2. Use a standard normal distribution table or a statistical software to find the probabilities associated with the z-scores.
P(69 ≤ X̄ ≤ 71) = P(z1 ≤ Z ≤ z2), where Z represents the z-scores for 69 and 71 inches.

Note: The steps provided here assume that the population distribution is approximately normal and the sample size is large enough for the Central Limit Theorem to hold. If these conditions are not met, alternative methods may be required to estimate probabilities or the sampling distribution.