The National Health Statistics Reports dated Oct. 22,2008...

The National Health Statistics Reports dated Oct. $22,2008$ stated that for a sample size of 27718 -year-old American males, the sample mean waist circumference was 86.3 $\mathrm{cm} .$ A some-what complicated method was used to estimate various population percentiles, resulting in the following values: $\begin{array}{cccccc}{5 \mathrm{th}} & {10 \mathrm{th}} & {25 \mathrm{th}} & {50 \mathrm{th}} & {75 \mathrm{th}} & {90 \mathrm{th}} & {95 \mathrm{th}} \\ {69.6} & {70.9} & {75.2} & {81.3} & {95.4} & {107.1} & {116.4}\end{array}$ (a) Is it plausible that the waist size distribution is at least approximately normal? Explain your reasoning. If your answer is no, conjecture the shape of the population distribution. (b) Suppose that the population mean waist size is 85 $\mathrm{cm}$ and that the population standard deviation is 15 $\mathrm{cm} .$ How likely is it that a random sample of 277 individuals will result in a sample mean waist size of at least 86.3 $\mathrm{cm} ?$ (c) Referring back to (b), suppose now that the population mean waist size is 82 $\mathrm{cm}$ (closer to the sample median than the sample mean). Now what is the (approximate) probability that the sample mean will be at least 86.3$?$ In light of this calculation, do you think that 82 is a reasonable value for $\mu ?$

The National Health Statistics Reports dated Oct. $22,2008$ stated that for a sample size
of 27718 -year-old American males, the sample mean waist circumference was 86.3 $\mathrm{cm} .$ A some-what complicated method was used to estimate various population percentiles, resulting in the
following values:
$\begin{array}{cccccc}{5 \mathrm{th}} & {10 \mathrm{th}} & {25 \mathrm{th}} & {50 \mathrm{th}} & {75 \mathrm{th}} & {90 \mathrm{th}} & {95 \mathrm{th}} \\ {69.6} & {70.9} & {75.2} & {81.3} & {95.4} & {107.1} & {116.4}\end{array}$
(a) Is it plausible that the waist size distribution is at least approximately normal? Explain your
reasoning. If your answer is no, conjecture the shape of the population distribution.
(b) Suppose that the population mean waist size is 85 $\mathrm{cm}$ and that the population standard deviation is 15 $\mathrm{cm} .$ How likely is it that a random sample of 277 individuals will result in a sample mean waist size of at least 86.3 $\mathrm{cm} ?$
(c) Referring back to (b), suppose now that the population mean waist size is 82 $\mathrm{cm}$ (closer to the sample median than the sample mean). Now what is the (approximate) probability that
the sample mean will be at least 86.3$?$ In light of this calculation, do you think that 82 is a
reasonable value for $\mu ?$

Key Concepts

Normal Distribution

The normal distribution is a symmetric, bell-shaped probability model defined by its mean and standard deviation. It has the property that about 68% of values fall within one standard deviation of the mean, and its percentiles are symmetrically arranged. This concept is fundamental in assessing whether observed data, such as sample percentiles, are consistent with what is expected from a normal distribution.

Skewness and Distribution Shape

Skewness refers to the asymmetry in the distribution of data around its mean. When comparing percentiles, if the spread below and above the median differs, the distribution might be skewed. Understanding skewness is essential when evaluating whether a distribution deviates from normality and determining if it is to the left (negative skew) or to the right (positive skew).

Central Limit Theorem

The Central Limit Theorem states that irrespective of the underlying population distribution, the distribution of the sample mean will tend to be normal if the sample size is sufficiently large. This concept is crucial when using sample means to make inferences about the population, especially in hypothesis testing and probability calculations involving the sample mean.

Sampling Distribution of the Mean

This concept involves the probability distribution of the means obtained from multiple samples drawn from the same population. It has a mean equal to the population mean and a standard error equal to the population standard deviation divided by the square root of the sample size. This principle is key in determining the probability of obtaining a sample mean at least as extreme as the one observed given a specific population mean.

Probability Calculations and Hypothesis Testing

Probability calculations involve quantifying the chance of observing a particular sample result under a given population parameter. Such calculations often use Z-scores derived from the sampling distribution of the mean and the corresponding cumulative distribution function (CDF). This process underlies hypothesis testing, where the probability of obtaining a sample mean as extreme as the observed one is used to evaluate the reasonableness of the presumed population mean.

Transcript

00:01 So for this problem, we're looking at some population percentiles, and we have some multifaceted.

00:07 It's a multifaceted question.

00:09 So we need to look at what's going on.

00:12 We know our sample size equals n.

00:15 And that's pretty much all we need right now.

00:17 We know that our standard deviation is 15 as well.

00:21 So for problem a, it is asking us, can we tell if the approximation is normal? so if we were to plot this on the line, they would want to see some type of symmetry between 60 and 120.

00:40 And when we plot this out, we end up being right here.

00:45 And i know that it may look normal, but it's called right skewed.

00:52 And the way we figure that out mathematically is we do the standard deviation and then we do over in squared, which is our population squared.

01:10 1527 equals 0 .91 which means that it is it's not it's an error given so that's how we find it is right skewed for the next problem and what that looks like on this is kind of what we're looking at we're going to use this for problem too so for the second problem we suppose that the population mean of the waist size is 85 centimeters and the deviation is 15.

01:53 So what we're going to do is if we have n is equal to 277 like we said and mu, which is our mean, is 85.

02:06 And let's do this in a different color.

02:08 So let's recap.

02:09 N is 277.

02:12 Our mean is 85 and our standard deviation is 15.

02:35 So how do we find the standard? no, i'm sorry, how do we find the probability that it will, the waste will be at least 86 .3? so what we're going to do is, is we need to look at a z score.

02:57 So how we do the z score is we do x minus omega, which is our mean over the variance.

03:12 And for this problem, and for this, this is equivalent to the variance.

03:24 So what we're going to do is it is x minus h.

03:31 And then we have the variance...

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