Waist size revisited Researchers measured the Waist Sizes of...

Waist size revisited Researchers measured the Waist Sizes of 250 men in a study on body fat. The true mean and standard deviation of the Waist Sizes for the 250 men are 36.33 in and 4.019 inches, respectively. In Exercise 35 you looked at the histograms of simulations that drew samples of sizes 2, 5, 10, and 20 (with replacement). The summary statistics for these simulations were as follows: $$\begin{array}{|c|c|c|}\hline n & \text { mean } & {\text { st. dev. }} \\ \hline 2 & {36.314} & {2.855} \\ {5} & {36.314} & {1.805} \\ {10} & {36.341} & {1.276} \\ {20} & {36.339} & {0.895}\\ \hline \end{array}$$ a) According to the Central Limit Theorem, what should the theoretical mean and standard deviation be for each of these sample sizes? b) How close are the theoretical values to what was observed in the simulation? c) Looking at the histograms in Exercise 35, at what sample size would you be comfortable using the Normal model as an approximation for the sampling distribution? d) What about the shape of the distribution of Waist Size explains your choice of sample size in part c?

Waist size revisited Researchers measured the Waist Sizes of 250 men in a study on body fat. The true mean and standard deviation of the Waist Sizes for the 250 men are 36.33 in and 4.019 inches, respectively. In Exercise 35 you looked at the histograms of simulations that drew samples of sizes 2, 5, 10, and 20 (with replacement). The summary statistics for these simulations were as follows:
$$\begin{array}{|c|c|c|}\hline n & \text { mean } & {\text { st. dev. }} \\ \hline 2 & {36.314} & {2.855} \\ {5} & {36.314} & {1.805} \\ {10} & {36.341} & {1.276} \\ {20} & {36.339} & {0.895}\\ \hline \end{array}$$
a) According to the Central Limit Theorem, what should the theoretical mean and standard deviation be for each of these sample sizes?
b) How close are the theoretical values to what was observed in the simulation?
c) Looking at the histograms in Exercise 35, at what sample size would you be comfortable using the Normal model as an approximation for the sampling distribution?
d) What about the shape of the distribution of Waist Size explains your choice of sample size in part c?

Key Concepts

Central Limit Theorem

The Central Limit Theorem is a fundamental concept in statistics which states that, regardless of the underlying population distribution, the distribution of the sample means will tend to be normal (or approximately normal) if the sample size is sufficiently large. This theorem also provides that the mean of the sampling distribution will equal the true population mean, and the standard deviation will be the population standard deviation divided by the square root of the sample size, a measure known as the standard error.

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean refers to the probability distribution of sample means computed from multiple samples drawn from the same population. This distribution has its own mean and standard deviation. According to the Central Limit Theorem, its mean equals the population mean and its standard deviation, often termed the standard error, decreases as the sample size increases, reflecting more precise estimates of the population mean.

Standard Error

The standard error is the standard deviation of the sampling distribution of a statistic, most frequently the sample mean. It quantifies the variability of the sample mean from sample to sample. Given a population standard deviation, the standard error is calculated by dividing that standard deviation by the square root of the sample size, thus illustrating how larger sample sizes lead to a reduction in variability and more reliable estimates.

Normal Approximation

Normal approximation involves using the normal distribution to approximate the distribution of the sample statistic, such as the sample mean. This is especially valid under the conditions of the Central Limit Theorem when the sample size is sufficiently large, enabling statisticians to make probability-based inferences even if the underlying population distribution is not normal.

Impact of Population Distribution Shape

The shape of the population distribution plays a crucial role in determining how quickly the sampling distribution of the sample mean approaches normality. If the underlying population is highly skewed or exhibits other non-normal features, larger sample sizes might be necessary for the sampling distribution to approximate normality satisfactorily, as suggested by the Central Limit Theorem. Conversely, if the population distribution is already roughly symmetric and bell-shaped, even smaller samples can yield a sampling distribution that is approximately normal.

Transcript

00:01 So researchers measured the weight size of 250 men in a study on body fats and the sample size is 250 men.

00:14 And the mean and the standard deviation is, so the true mean is 36 .33 and then the standard deviation is 4 .019 inches.

00:34 So in this exercise you looked at the histograms of simulations that drew.

00:43 So basically the summary statistics are given in the following with the mean, the n and the standard deviation.

00:52 And the first question wants us to find, it says according to the central limit theorem, what should be the theoretical mean and standard deviation for each of the sample sizes? so the waist size a studied um study measured the weight size of so 50 men that had a mean of 36 .3 and a standard deviation of 4 .2 um inches approximately so we can draw a histogram of that measurement and that should look like this with weight size of 3 30 to 40 or we don't have to draw it because the question doesn't require so to explore how the the mean my very fun sample to sample they simulated by drawing many samples of two five ten and twenty now from the two fifty measurements so moving on according to the central limit theorem if assumptions of independence and random sampling are met and the sample size is large enough then the sampling distribution of the sample mean is model by model with the mean equals to the population mean so according to the theorem if the requirements are met then the mean will be equals to the population mean and the population mean and this is the formula the and then standard deviation will be equals to square root of n so the theoretical mean is this in essence and standard deviation is s d equals to this so for all the sample sizes the theoretical mean be equals to this equals to 36 .33 we're already giving that in the question if the requirements are meant then to be 36 .33 and the standard deviation will be this over square root of n which is equal to 4 .019 divided by the square root of n so that's the answer.

03:18 Now we can so moving on we're going to um show a table that shows the observance me this year it's coming the observance division and the theoretical standard division so this is the n this is the observed mean then we have um observed mean then theoretical mean then observe standard division then the theoretical standard division which is equals to the formula over square root of n so the n is 2 5 10 and 20 the observed means are the 36 .314, 36 .314, 36 .341, and 36 .339.

04:45 Now, the charity can mean is the same 36 .33, 36 .33, 36 .33, 36 .33, and 36 .33.

04:55 Now, the observed standard division is 2 .855, 1 .805, 1 .276...

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