Question

Putting It Together: Hand Washing The American Society for Microbiology (ASM) and the Soap and Detergent Association (SDA) jointly commissioned two separate studies during 2007 both of which were conducted by Harris Interactive. In one of the studies, 1001 adults were interviewed by telephone and asked about their hand-washing habits. In the other study, the hand washing behavior of 6076 adults was inconspicuously observed within public rest rooms in four U.S. cities. (a) In the telephone interviews, 921 of the adults said they always wash their hands in public rest rooms. Use this result to obtain a point estimate for the proportion of adults who say they always wash their hands in public rest rooms. (b) Verify that the requirements for constructing a confidence interval for $p$ from part (a) are satisfied. (c) Using the results from the telephone interviews, construct a $95 \%$ confidence interval for the proportion of adults who say they always wash their hands in public rest rooms. (d) Is it possible that the proportion of adults who say they always wash their hands in public rest rooms is less than $0.85 ?$ Is this likely? (e) In the observational study, 4679 of the 6076 adults were observed washing their hands. Use this result to obtain a point estimate for the proportion of adults who wash their hands in public rest rooms. (f) Verify that the requirements for constructing a confidence interval for $p$ from part (e) are satisfied. (g) Using the results from the observational study, construct a $95 \%$ confidence interval for the proportion of adults who wash their hands in public rest rooms. (h) Is it possible that the proportion of adults who wash their hands in public rest rooms is greater than $0.85 ?$ Is this likely? (i) Based on your findings in parts (a) through (h), what might you conclude about the proportion of adults who say they always wash their hands versus the proportion of adults who actually wash their hands in public rest rooms? Why do you think this is?

Name: Problem 45, Chapter 9: Statistics Informed Decisions Using Data
Uploaded: 2021-07-01T03:24:25-08:00
Duration: 12 min 24 s
Channel: Diane Koenig

   Putting It Together: Hand Washing The American Society for Microbiology (ASM) and the Soap and Detergent Association (SDA) jointly commissioned two separate studies during 2007 both of which were conducted by Harris Interactive. In one of the studies, 1001 adults were interviewed by telephone and asked about their hand-washing habits. In the other study, the hand washing behavior of 6076 adults was inconspicuously observed within public rest rooms in four U.S. cities.
(a) In the telephone interviews, 921 of the adults said they always wash their hands in public rest rooms. Use this result to obtain
a point estimate for the proportion of adults who say they always wash their hands in public rest rooms.
(b) Verify that the requirements for constructing a confidence interval for $p$ from part (a) are satisfied.
(c) Using the results from the telephone interviews, construct a $95 \%$ confidence interval for the proportion of adults who say they always wash their hands in public rest rooms.
(d) Is it possible that the proportion of adults who say they always wash their hands in public rest rooms is less than $0.85 ?$ Is this likely?
(e) In the observational study, 4679 of the 6076 adults were observed washing their hands. Use this result to obtain a point estimate for the proportion of adults who wash their hands in public rest rooms.
(f) Verify that the requirements for constructing a confidence interval for $p$ from part (e) are satisfied.
(g) Using the results from the observational study, construct a $95 \%$ confidence interval for the proportion of adults who wash their hands in public rest rooms.
(h) Is it possible that the proportion of adults who wash their hands in public rest rooms is greater than $0.85 ?$ Is this likely?
(i) Based on your findings in parts (a) through (h), what might you conclude about the proportion of adults who say they always wash their hands versus the proportion of adults who actually wash their hands in public rest rooms? Why do you think this is?

Statistics Informed Decisions Using Data

Michael Sullivan III 4th Edition

Chapter 9, Problem 45

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Solved by Expert Diane Koenig

07/01/2021

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The point estimate for the proportion of adults who say they always wash their hands in public rest rooms is given by the ratio of the number of adults who said they always wash their hands to the total number of adults interviewed. This is calculated as Show more…

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Putting It Together: Hand Washing The American Society for Microbiology (ASM) and the Soap and Detergent Association (SDA) jointly commissioned two separate studies during 2007 both of which were conducted by Harris Interactive. In one of the studies, 1001 adults were interviewed by telephone and asked about their hand-washing habits. In the other study, the hand washing behavior of 6076 adults was inconspicuously observed within public rest rooms in four U.S. cities. (a) In the telephone interviews, 921 of the adults said they always wash their hands in public rest rooms. Use this result to obtain a point estimate for the proportion of adults who say they always wash their hands in public rest rooms. (b) Verify that the requirements for constructing a confidence interval for $p$ from part (a) are satisfied. (c) Using the results from the telephone interviews, construct a $95 \%$ confidence interval for the proportion of adults who say they always wash their hands in public rest rooms. (d) Is it possible that the proportion of adults who say they always wash their hands in public rest rooms is less than $0.85 ?$ Is this likely? (e) In the observational study, 4679 of the 6076 adults were observed washing their hands. Use this result to obtain a point estimate for the proportion of adults who wash their hands in public rest rooms. (f) Verify that the requirements for constructing a confidence interval for $p$ from part (e) are satisfied. (g) Using the results from the observational study, construct a $95 \%$ confidence interval for the proportion of adults who wash their hands in public rest rooms. (h) Is it possible that the proportion of adults who wash their hands in public rest rooms is greater than $0.85 ?$ Is this likely? (i) Based on your findings in parts (a) through (h), what might you conclude about the proportion of adults who say they always wash their hands versus the proportion of adults who actually wash their hands in public rest rooms? Why do you think this is?

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Key Concepts

Comparison of Self-Reported and Observed Behaviors

This concept highlights the differences that can arise between behaviors reported in surveys and those directly observed. Self-reported data can be influenced by biases such as social desirability bias, where individuals tend to overstate behaviors that are viewed favorably. In contrast, observational data tends to provide a more objective measure of actual behavior, offering insights into discrepancies between what people say they do and what they actually do.

Normal Approximation Conditions for Proportions

When constructing confidence intervals for proportions, the normal approximation is often used if the sample size is sufficiently large. Specifically, the conditions require that both np and n(1 - p) are at least 10. This ensures that the distribution of the sample proportion can be well-approximated by a normal distribution, which justifies using z-scores in the calculation of the interval.

Point Estimate for a Proportion

This concept involves using sample data to estimate the true proportion of a population that exhibits a certain characteristic. It is calculated as the number of individuals showing the behavior divided by the total number of individuals in the sample. The point estimate serves as a single best guess of the population proportion based on the observed data.

Confidence Interval for a Proportion

A confidence interval gives a range of plausible values for the true population proportion. It is constructed from the sample proportion, incorporating a margin of error which is determined using the standard error and a critical value from the normal distribution corresponding to a desired confidence level, such as 95%. This interval helps to account for sampling variability, providing insight into the precision and reliability of the point estimate.

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Transcript

00:01 With this example, we're going to look at a couple different settings where we've actually taken a sample of one group and found the number that have that characteristic in that first group.

00:12 And then we do it for a second group as well.

00:15 So here for our first group, our sample size is 1001.

00:18 And the number in the sample that have the characteristic we're focusing on is 921.

00:23 For a point estimate p hat, we find it by x over n.

00:29 So 921 divided by 10001, or we get our p hat value of 0 .92.

00:39 Now we want to see if the conditions are met that we can use the confidence interval techniques from this presentation of the material.

00:48 So is n times p hat times 1 minus p hat greater than or equal to 10? will calculate out the 1001 times a 0 .92 times the 1 minus 0 .92.

01:03 And when you do get that, you'll get a 73 .67, which definitely is greater than 10.

01:13 So that's yes.

01:15 Is the sample size less than equal to 5 % of the population size? well, if this is from adult americans, we want to know if 100 % is less than 1 ,000 is less or equal to 0 .05 is the decimal form for 5%.

01:31 And the number of adult americans at the most recent census is 257 ,536 ,000.

01:41 And so definitely 1001 is smaller than that number.

01:49 So that's also meant.

01:52 Now, what if they want us to then construct a 95 % confidence interval for this population proportion p.

02:13 We'll remember it's p hat minus e for a lower bound, and it's p hat plus e for an upper bound, where e is equal to z sub alpha over 2 times the square root of p hat times one minus p hat over n, and if it's a 90 % confidence level, your z sub alpha over 2 is the 1 .645.

02:48 If it's a 95 % confidence level, z sub alpha over 2 is 1 .960.

02:56 And if it's a 99 % confidence level, your z sub alpha over 2 is 2 .575.

03:04 So those are three different ones that it's always good to have reference or commit to memory because they're used quite frequently.

03:12 There is an earlier video that talks about how these critical values are found.

03:17 Also, it walks you through if you have a different percentage than these, like if you want it for an 85 % or anything like that.

03:25 But here, it's for 95 % confidence interval.

03:29 So we're going to use our z -sive alpha over 2 of the 1 .9.

03:34 960.

03:36 So we have our p hat, which we found in part a.

03:41 So that's the 0 .920 minus, and then our e calculation is a 1 .960 times the square root of our p hat of 0 .920 times 1 minus 0 .920 over our n.

04:04 And n is, and n is 1 ,001 and then that's our lower bound our upper bound is 0 .920 plus that 1 .960 times that square root of 0 .92 times 1 minus 0 .92 divided by the 1001 so calculating that out you get that our lower bound is 0 .903 and our upper bound upper bound is 0 .937.

04:42 Now what if it asks me, is it possible that the population proportion is less than 0 .85? well, 0 .85 is not in my confidence interval, but this is a 95 % confidence...

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