How tall? The heights of young men follow a Normal...

How tall? The heights of young men follow a Normal distribution with mean 69.3 inches and standard deviation 2.8 inches. The heights of young women follow a Normal distribution with mean 64.5 inches and standard deviation 2.5 inches. Suppose we select independent SRSs of 16 young men and 9 young women and calculate the sample mean heights $\bar{x}_{M}$ and $\bar{x}_{W}$. (a) What is the shape of the sampling distribution of $\bar{x}_{M}-\bar{x}_{W} ?$ Why? (b) Find the mean of the sampling distribution. Show your work. (c) Find the standard deviation of the sampling distribution. Show your work.

How tall? The heights of young men follow a Normal distribution with mean 69.3 inches and standard deviation 2.8 inches. The heights of young women follow a Normal distribution with mean 64.5 inches and standard deviation 2.5 inches. Suppose we select independent SRSs of 16 young men and 9 young women and calculate the sample mean heights $\bar{x}_{M}$ and $\bar{x}_{W}$.
(a) What is the shape of the sampling distribution of $\bar{x}_{M}-\bar{x}_{W} ?$ Why?
(b) Find the mean of the sampling distribution. Show your work.
(c) Find the standard deviation of the sampling distribution. Show your work.

Key Concepts

Sample Mean

The sample mean is the arithmetic average of observations in a sample and serves as an unbiased estimator of the population mean. When the data come from a normal distribution, the sample mean itself is normally distributed, which simplifies inference and hypothesis testing.

Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric around its mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. It is defined by its mean and standard deviation and forms the familiar bell-shaped curve used to model many natural phenomena.

Sampling Distribution

The sampling distribution is the probability distribution of a given statistic based on a random sample. It represents how the statistic varies from sample to sample and is fundamental in making inferences about the population from which the sample is drawn.

Difference of Two Independent Sample Means

When two independent sample means are computed from separate randomly selected samples, the difference between them is itself a random variable. If the original populations are normally distributed, then this difference will also follow a normal distribution, with its mean equal to the difference of the population means and its variance being a sum of the individual variances divided by their respective sample sizes.

Standard Error

The standard error measures the variability or dispersion of a sampling distribution by quantifying how much the observed sample statistic is expected to vary from the true population parameter. For a sample mean, it is calculated by dividing the population standard deviation by the square root of the sample size, and for the difference of two independent sample means, the variances of the two means are added together (assuming independence) before taking the square root.

Transcript

00:01 In this video, we are going to focus on finding the shape of the sampling distribution, mean of the sampling distribution, as well as standard deviation of the sampling distribution based on the given information.

00:11 So here the sample size for the first sample is 16, then the sample size for the second sample is 9, and then here we can write that mu 1 value, that is mean for first sample is 69 .3, and then mu 2 value will be here equals to 64 .5.

00:30 Then we can say that the value of sigma 1, that is standard deviation will be here, 2 .8, and then here we can say that value of sigma 2 will be equals to 2 .5.

00:40 So we have been given that the population distribution of the first normal, sorry, first population is normal and the population distribution of second population is also normal distribution.

00:51 Since both, let's write over here that both distributions, d -i -t -r -i, b -u -t -i -o -n -s are normal.

01:04 So, the shape of the sampling distribution will also be normal.

01:09 So here we can write that shape will be equals to, so we are talking about the shape of x -m -m -minus, it is x -w, right? and it can equivalently return as, here it will be x -1 -1 -6 -2 -bar.

01:26 So it is x1 bar minus x2 bar basically.

01:30 And then this very value, we can say that the shape is here equals to normal for the sampling distribution as well.

01:38 And this is the answer for the first part of this very problem.

01:41 And we can put this thing inside a box and it will look like this...

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