Serum Cholesterol As reported by the U.S. National Center...

Serum Cholesterol As reported by the U.S. National Center for Health Statistics, the mean serum high density lipoprotein (HDL) cholesterol of males 20 to 29 years old is $\mu=47 .$ Dr. Paul Oswiecmiski wants to estimate the mean serum HDL cholesterol of his $20-$ to 29 -year-old male patients. He randomly selects 15 of his 20 - to 29 -yearold patients and obtains the data shown. Assume that $\sigma=12.5$ (TABLE CAN'T COPY) (a) Use the data to compute a point estimate for the population mean serum HDL cholesterol in Dr. Oswiecmiski's patients. (b) An analysis of the data indicate that the sample data come from a population that is approximately normalIy distributed with no outliers. Construct a $95 \%$ confidence interval for the mean serum HDL cholesterol for all Dr. Oswiecmiski's $20-$ to 29 -year-old male patients. Interpret this interval. (c) Do Dr. Oswiecmiski's patients appear to have a serum HDL different from that of the general population. (d) If Dr. Oswiecmiski wanted a more precise confidence interval, what would you recommend?

Serum Cholesterol As reported by the U.S. National Center for Health Statistics, the mean serum high density lipoprotein (HDL) cholesterol of males 20 to 29 years old is $\mu=47 .$ Dr. Paul Oswiecmiski wants to estimate the mean serum HDL cholesterol of his $20-$ to 29 -year-old male patients. He randomly selects 15 of his 20 - to 29 -yearold patients and obtains the data shown. Assume that $\sigma=12.5$ (TABLE CAN'T COPY)
(a) Use the data to compute a point estimate for the population mean serum HDL cholesterol in Dr. Oswiecmiski's patients.
(b) An analysis of the data indicate that the sample data come from a population that is approximately normalIy distributed with no outliers. Construct a $95 \%$ confidence interval for the mean serum HDL cholesterol for all Dr. Oswiecmiski's $20-$ to 29 -year-old male patients. Interpret this interval.
(c) Do Dr. Oswiecmiski's patients appear to have a serum HDL different from that of the general population.
(d) If Dr. Oswiecmiski wanted a more precise confidence interval, what would you recommend?

Key Concepts

Population Mean and Point Estimation

The population mean (?) is the true average value of the characteristic in the entire population. A point estimate, typically the sample mean, is used to approximate this value based on a sample drawn from the population. This estimate serves as the best single value predictor of the overall population parameter, acknowledging that it may differ from the true mean due to sampling variability.

Confidence Interval

A confidence interval provides a range of values around a point estimate that is likely to contain the true population mean with a certain level of confidence (e.g., 95%). When the population standard deviation is known and the data are approximately normally distributed, a z-statistic is used to account for sampling variability. This interval offers a quantifiable measure of uncertainty associated with the point estimate.

Sampling Distribution and Central Limit Theorem

The sampling distribution of the sample mean refers to the probability distribution of all possible sample means that could be obtained from the population. According to the Central Limit Theorem, as long as the sample size is sufficiently large, the distribution of the sample mean will be approximately normal, regardless of the population distribution. This principle underpins the validity of using confidence intervals and hypothesis tests for the mean.

Hypothesis Testing for Means

Hypothesis testing for means involves comparing a sample-based estimate to a hypothesized population mean to determine if there is a statistically significant difference. In this context, the test usually establishes a null hypothesis that there is no difference between the sample mean and the population mean, and then uses a test statistic (often a z-score when the standard deviation is known) to assess if any observed difference is likely to be due to random sampling variability or represents a true difference.

Effect of Sample Size on Precision

The precision of an estimate is influenced by the sample size; larger samples tend to produce more precise (narrower) confidence intervals because they reduce the standard error of the estimate. Increasing the sample size decreases the variability due to randomness, thereby allowing for a more accurate estimation of the population parameter. This is a central concept in determining how to improve the reliability of statistical estimates.

Transcript

00:01 The following is a solution video.

00:01 It's number 29, and this looks at dr.

00:04 I'm going to butcher this name, but dr.

00:09 Oswik -miskees, 20 to 29 -year -old patients, their mean serum hdl cholesterol level.

00:16 And we're given a data set.

00:18 Now, it's hypothesized that the mean is 47 and that from previous studies the population standard deviation is 12 .5.

00:25 So we're assuming actually, in fact, it even says as we're assuming normality here.

00:29 So we can use the z interval because we know the population standard deviation.

00:33 We actually know what sigma is.

00:35 Whenever you know what sigma is, you can use the z interval.

00:37 So the first part is to find the point estimate.

00:39 Now i use technology to do that, but you can use the formula if you so wish.

00:42 But if you have a ti .84 and 83, you go to stat and then edit.

00:47 And you can see here are the data values here.

00:49 So i don't know if you saw all those, but there should be 15 data values.

00:53 I think there are 15.

00:54 Let me just double check real quick.

00:55 Yeah, 15 data values.

00:58 And if you go to stat again, and then arrow over to calc, and then you click on one -var stats.

01:02 Now, i put my list in l -1, so i'm going to keep that as l -1, and then you press calculate, and that x -bar, that's your point estimate for the means, so 48 .33, for you repeating, i guess, 48 and a third.

01:13 So 48, we'll just say 48 .3 is the point estimate.

01:19 Okay, so now we've got to find the 95 % confidence intervals.

01:22 So again, i'm going to turn to the calculator because it's a lot quicker this way, but you can certainly use the formula.

01:26 And we go to tests, and it's the seventh option here.

01:30 Like i said, we can use the z interval whenever we know what sigma is.

01:35 And we are working in data here, so we're going to keep the data up here highlighted for input.

01:40 So it's not summary stats.

01:41 It's actual data set.

01:42 The sigma, the standard deviation was 12 .5.

01:45 The list was l1.

01:46 At least mine was, i put it in l1.

01:48 Frequency should be one.