In a survey funded by a biomedical research foundation, 750...

In a survey funded by a biomedical research foundation, 750 of 1000 adult Americans said they didn't believe they could come down with a sexually transmitted disease (STD). Construct a $95 \%$ confidence interval estimate of the proportion of all adult Americans who don't believe they can contract an STD. (A) $0.750 \pm 1.645 \sqrt{1000(0.750)(0.250)}$ (B) $0.750 \pm 1.645 \sqrt{\frac{(0.750)(0.250)}{1000}}$ (C) $0.750 \pm 1.96 \sqrt{1000(0.750)(0.250)}$ (D) $0.750 \pm 1.96 \sqrt{\frac{(0.750)(0.250)}{1000}}$ (E) $0.750 \pm 2.576 \sqrt{1000(0.750)(0.250)}$

In a survey funded by a biomedical research foundation, 750 of 1000 adult Americans said they didn't believe they could come down with a sexually transmitted disease (STD). Construct a $95 \%$ confidence interval estimate of the proportion of all adult Americans who don't believe they can contract an STD.
(A) $0.750 \pm 1.645 \sqrt{1000(0.750)(0.250)}$
(B) $0.750 \pm 1.645 \sqrt{\frac{(0.750)(0.250)}{1000}}$
(C) $0.750 \pm 1.96 \sqrt{1000(0.750)(0.250)}$
(D) $0.750 \pm 1.96 \sqrt{\frac{(0.750)(0.250)}{1000}}$
(E) $0.750 \pm 2.576 \sqrt{1000(0.750)(0.250)}$

Key Concepts

Critical Value (z-score) Selection

The critical value, or z-score, corresponds to the desired level of confidence for the interval estimate. For a 95% confidence interval, the standard z-score is approximately 1.96. This value is multiplied by the standard error to determine the margin of error in the confidence interval formula (p ± z*?(p(1-p)/n)).

Standard Error of a Proportion

The standard error for a proportion quantifies the variability of the sampling distribution of the sample proportion. It is calculated using the formula ?(p(1-p)/n), where p is the sample proportion and n is the sample size. This value is critical in constructing the confidence interval, as it represents the estimated standard deviation of the sample proportion.

Confidence Interval for a Population Proportion

This concept involves estimating a range within which the true population proportion will lie with a certain level of confidence. The confidence interval is typically constructed using the sample proportion, a critical value from the normal distribution (based on the desired confidence level), and the standard error of the proportion. It provides a measure of uncertainty around the sample estimate.

Transcript

00:01 Number four, in a survey funded by a biomedical research foundation, 750 of 1 ,000 adult americans said they didn't believe they could come down with sexually transmitted disease.

00:12 Construct a 95 % confidence interval estimate of the proportion of all adult americans who don't believe they can contract an std.

00:19 So we have our sample proportion to be 750 over 1 ,000, which is 0 .75.

00:29 We want to construct a 95 % confidence interval.

00:35 So look at all the answer choices.

00:37 We don't really have confidence intervals solved out.

00:40 We have them set up.

00:41 So we're going to set ours up.

00:43 If we're looking for a confidence interval, we need p -hat plus or minus critical value times our standard error of p -hat, which we can write out p -hat, 1 -p -hat over in.

01:03 Okay, plugging in what we know, we know that p hat is 0 .75.

01:07 We'll come back to z -a -strix here in just a second.

01:11 Because we know p -had is 0 .75, we can plug that in as well.

01:16 Sample size is 1 ,000.

01:19 So the first thing i'm going to do before i find z -astrics, because this is a multiple -choice question, is eliminate anything that doesn't match what i currently have.

01:30 Well, i have 1 ,000 as my denominator under my radical, a can't be it.

01:37 So we can eliminate a.

01:38 We can also eliminate c, and we can eliminate e.

01:41 So i'm down to b or d...

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