Probability Calculations for Means (Class)

The Central Limit Theorem

The Central Limit Theorem states that for a large enough sample size (\(n>30\)) the distribution of sample means is approximately normal with mean, \(\mu_{\bar{x}} = \mu\) and \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\) regardless of the distribution of the population.

We can assume the distribution of sample means is approximately normal if:

The population is normally distributed
n > 30

Don’t forget, that if the population is normally distributed, so is the distribution of sample means regardless of sample size.

Probability Calculations for the Sampling Distribution of \(\bar{x}\)

When we are confident that the sampling distribution of \(\bar{x}\) is normal, we can use the pnorm() function just as we did before. We simply input the mean, \(\mu_{\bar{x}} = \mu\) and \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\) into the Z-score formula as follows:

xbar <- 13.5
mu <- 20
sigma <- 15
n <- 33
sigma_xbar <- sigma / sqrt(n)

z <- (xbar - mu) / (sigma_xbar)
z

[1] -2.48931

# Left tail [Prob(some sample mean LESS THAN xbar)]
pnorm(z)

[1] 0.006399557

# Right tail [Prob(some sample mean GREATER THAN xbar)]
1-pnorm(z)

[1] 0.9936004

# The probability of a sample mean being between A and B

A <- 9
B <- 15
pnorm(B, mu, sigma_xbar) - pnorm(A, mu, sigma_xbar)

[1] 0.02774293

# Calculating Percentiles: What is the sample mean below which a certain percent is found? 

qnorm(.25, mu, sigma_xbar)

[1] 18.2388

One Z Formula to Rule Them All

We do not have to worry about choosing between two different Z-score formulas (one for individuals and a separate one for means).

If we are calculating probabilities for a normal population and we want to calculate the probability of a single individual, we can simply set \(n=1\).

WARNING: Using sample size of 1 only works if the population is already normally distributed. But if we confident that it is, we can use the following calculations:

xbar <- 13.5
mu <- 20
sigma <- 15
sigma_xbar <- sigma / sqrt(n)
n <- 1

z <- (xbar - mu) / (sigma_xbar)
z

[1] -2.48931

# Left tail [Prob(some sample mean LESS THAN xbar)]
pnorm(z)

[1] 0.006399557

# Right tail [Prob(some sample mean GREATER THAN xbar)]
1-pnorm(z)

[1] 0.9936004

Practice Together

Use the following calculator to answer the questions below.

REMEMBER: Whenever you update information, it is useful to run the whole code chunk.

GPA’s

Suppose the mean GPA of BYU-Idaho students is 3.5 and the standard deviation is 0.7. It is well known that this distribution is left-skewed. A random sample of n = 45 students will be drawn.

Question: What is the mean of the distribution of the sample means (sampling distribution) for all possible samples of size 45 that could be drawn from the parent population of GPAs?
Answer:

Question: What is the standard deviation of the distribution of the sample means (sampling distribution) for all possible samples of size 45 that could be drawn from the parent population?
Answer:

Question: What is the probability that the mean GPA for 45 randomly selected BYU-Idaho students will be less than 3.3?
Answer:

Question: What is the shape of the distribution of sample means, \(\bar{x}\), when 45 students are selected?
Answer:

GRE Scores

Scores on the quantitative portion of the GRE are approximately normally distributed with mean, \(\mu=150.8\), and standard deviation \(\sigma = 8.8\).

Question: Dianne earned a score of 160 on the quantitative portion of the GRE. What is the z-score corresponding to Dianne’s score?
Answer:

Question: What is the probability that a randomly selected student will score above 160 on the quantitative portion of the GRE?
Answer:

Question: What is the probability that a randomly selected group of 18 students will have an average less than 160 on the quantitative portion of the GRE?
Answer:

Question: What is the probability that a randomly selected group of 18 students will have an average between 145 and 160 on the quantitative portion of the GRE?
Answer: