Sexual Relations A question on the 2006 General Social...

Sexual Relations A question on the 2006 General Social Survey was this: "About how many times did you engage in intercourse during the month?" The question was only asked of participants who had previously indicated that they had engaged in sexual intercourse during the past month. The results are summarized in the MINITAB output that follows: One-Sample T: SEXFREQ2 Variable $\quad$ N Mean StDev SB Mean 95.03 CI SBXFREQ2 357 7.678 6.843 $0.362 \quad(6.966,8.390)$ (a) Use the summary to determine the point estimate of the population mean and margin of error for the confidence interval. (b) Interpret the confidence interval. (c) Verify the results by computing a $95 \%$ confidence interval with the information provided.

Sexual Relations A question on the 2006 General Social Survey was this: "About how many times did you engage in intercourse during the month?" The question was only asked of participants who had previously indicated that they had engaged in sexual intercourse during the past month. The results are summarized in the MINITAB output that follows:
One-Sample T: SEXFREQ2 Variable $\quad$ N Mean StDev SB Mean 95.03 CI SBXFREQ2 357 7.678 6.843
$0.362 \quad(6.966,8.390)$
(a) Use the summary to determine the point estimate of the population mean and margin of error for the confidence interval.
(b) Interpret the confidence interval.
(c) Verify the results by computing a $95 \%$ confidence interval with the information provided.

Key Concepts

Point Estimate

A point estimate is a single numerical value calculated from sample data that serves as the best estimate of a corresponding population parameter. For example, when estimating a population mean, the sample mean is used as the point estimate.

Margin of Error

The margin of error represents the range of uncertainty around the point estimate and indicates how close the true population parameter is expected to lie relative to the estimate. It is calculated as the product of a critical value (from a relevant probability distribution) and the standard error of the estimate.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true value of the population parameter with a specified level of confidence (such as 95%). It combines the point estimate and the margin of error to provide a measure of estimate precision.

t-Distribution

The t-distribution is used in inference when the sample size is relatively small or when the population standard deviation is unknown. It adjusts for the additional uncertainty of estimating variability from the sample data and is central to constructing confidence intervals for the mean.

Transcript

00:03 In this video, we're going to look at a mini tab output.

00:06 Minitab is a statistical software package that many schools have their students use in the statistical courses that they're taking, and also many businesses use when they're doing statistical analysis for their applications.

00:24 And with the mini tab output, the main thing is just learning how to read the table of information it gives you and how to interpret the results.

00:34 So here for our mini -tap output, the program was run with the one -sample -t test, and the variable is the sex frequency.

00:47 Your sample size n is 357, and normally in statistics we use small -case n for our sample size and capital n for the population size, just in terms of the readout for minitab, even though it says capital n, that is my sample size of 357.

01:06 The mean is 7 .678.

01:11 The standard deviation is 6 .843.

01:15 Standard error of the mean is 0 .362.

01:19 And my 95 % confidence interval has a lower bound of 6 .966 and an upper bound of 8 .6 .6 .6.

01:30 0 .390.

01:33 So that just kind of overviews what we have for the readout.

01:37 And now let's answer the specific question.

01:40 So what is the point estimate of the population mean? and remember, mu is the symbol for the population mean.

01:47 So that could be worded out as the population mean, or they could actually just say mu.

01:52 And it is portraying to you the same question.

01:56 Well, the point estimate of your population mean is your sample mean.

02:02 And in this readout, the sample mean is the 7 .678.

02:13 How about for the margin of error? well, the margin of error isn't specifically given to you here, but it's embedded within the process.

02:22 Just kind of recall that when we made our confidence intervals for our population mean, our lower bound was the x bar minus the t sub alpha over two, times s over the square root of n.

02:40 And the upper bound was x bar plus t sub alpha over two times s over the square root of n.

02:49 And this t sub alpha over two times s over the square of n, that is your margin of air.

02:58 That's your e.

03:00 So if you have this upper bound number and you subtract off the x bar, you're left with e.

03:11 So here, our upper bound is 8 .390 minus my mean is 7 .678.

03:26 And when i do that subtraction, i get a margin of error.

03:30 E equal to 0 .712.

03:39 Now, what's the next question? interpret the confidence interval.

03:44 So when we want to interpret the confidence interval, we need to state what our level of confidence is and what parameter we're talking about.

03:55 So here it was a 95 % confidence level, and it's about the population mean.

04:06 So we are 95 % confident...

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