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e. In constructing a confidence interval estimate of p, you have 850 survey responses and 10% of them answered "yes" to the first question. 5. Sample Size You have been hired by a college foundation to conduct a survey of graduates. a. If you want to estimate the percentage of graduates who have made a donation to the college after graduation, how many graduates must you survey if you want 98% confidence that your percentage has a margin of error of five percentage points? b. If you want to estimate the mean amount of all charitable contributions made by graduates, how many graduates must you survey if you want 98% confidence that your sample mean is in error by no more than $50? (Based on results from a pilot study, assume that the standard deviation of donations by graduates is $337.) If you plan to obtain the estimates described in parts (a) and (b) with a single survey having several questions, how many graduates must be surveyed? 6. Alcohol Consumption In a Gallup poll, 1011 adults were asked if they consume alcoholic beverages, and 64% of them said that they did. Construct a 90% confidence estimate of the proportion of all adults who consume alcoholic beverages. Can we conclude that the majority of adults consume alcoholic beverages? 7. Wristwatch Accuracy Students of the author collected data measuring the accuracy of wristwatches. The times (sec) below show the discrepancy between the real time and the time indicated on the wristwatch. Negative values correspond to watches that are running ahead of the actual time. The data satisfy a loose requirement of appearing to come from a normally distributed population. Construct a 95% confidence interval estimate of the mean discrepancy for the population of wristwatches. -85 325 20 305 -93 15 282 27 555 570 -241 36 8. White Blood Cell Counts Data Set 1 in Appendix B lists the white blood cell counts (1000 cells/?L) of 40 randomly selected women. The mean of those 40 values is 7.15 and the standard deviation is 2.28. Construct a 90% confidence interval estimate of the mean white blood cell count for the population of women. Does the confidence interval also serve as an estimate of the mean white blood cell count of men? 9. Car Crash Tests Data Set 13 in Appendix B includes crash test measurements for small cars. The seven chest deceleration measurements have a mean of 42.7 g and a standard deviation of 5.6 g, where g is a force of gravity. Use the sample data to construct a 95% confidence interval estimate of the mean chest deceleration measurement for the population of all small cars. (Assume that the sample is a simple random sample and the measurements satisfy the loose requirement of being from a normally distributed population.) Write a brief statement that interprets the confidence interval. 10. Car Crash Tests Refer to the sample data described in Exercise 9 and construct a 95% confidence interval estimate of the population standard deviation.

e. In constructing a confidence interval estimate of p, you have 850 survey responses and 10% of them answered "yes" to the first question.
5. Sample Size You have been hired by a college foundation to conduct a survey of graduates.
a. If you want to estimate the percentage of graduates who have made a donation to the college after graduation, how many graduates must you survey if you want 98% confidence that your percentage has a margin of error of five percentage points?
b. If you want to estimate the mean amount of all charitable contributions made by graduates, how many graduates must you survey if you want 98% confidence that your sample mean is in error by no more than $50? (Based on results from a pilot study, assume that the standard deviation of donations by graduates is $337.)
If you plan to obtain the estimates described in parts (a) and (b) with a single survey having several questions, how many graduates must be surveyed?
6. Alcohol Consumption In a Gallup poll, 1011 adults were asked if they consume alcoholic beverages, and 64% of them said that they did. Construct a 90% confidence estimate of the proportion of all adults who consume alcoholic beverages. Can we conclude that the majority of adults consume alcoholic beverages?
7. Wristwatch Accuracy Students of the author collected data measuring the accuracy of wristwatches. The times (sec) below show the discrepancy between the real time and the time indicated on the wristwatch. Negative values correspond to watches that are running ahead of the actual time. The data satisfy a loose requirement of appearing to come from a normally distributed population. Construct a 95% confidence interval estimate of the mean discrepancy for the population of wristwatches.
-85 325 20 305 -93 15 282 27 555 570 -241 36
8. White Blood Cell Counts Data Set 1 in Appendix B lists the white blood cell counts (1000 cells/?L) of 40 randomly selected women. The mean of those 40 values is 7.15 and the standard deviation is 2.28. Construct a 90% confidence interval estimate of the mean white blood cell count for the population of women. Does the confidence interval also serve as an estimate of the mean white blood cell count of men?
9. Car Crash Tests Data Set 13 in Appendix B includes crash test measurements for small cars. The seven chest deceleration measurements have a mean of 42.7 g and a standard deviation of 5.6 g, where g is a force of gravity. Use the sample data to construct a 95% confidence interval estimate of the mean chest deceleration measurement for the population of all small cars. (Assume that the sample is a simple random sample and the measurements satisfy the loose requirement of being from a normally distributed population.) Write a brief statement that interprets the confidence interval.
10. Car Crash Tests Refer to the sample data described in Exercise 9 and construct a 95% confidence interval estimate of the population standard deviation.

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Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

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Solved by Expert Teresa Wray

06/28/2022

Step 1

Given: - Total survey responses: 850 - Number of responses answering "Yes": 1020 - Margin of error: 5 percentage points - Confidence level: 98% Using the formula for sample size calculation for proportions: \[ n = \left( \frac{Z^2 \cdot p \cdot (1-p)}{E^2} Show more…

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In constructing a confidence interval estimate of p, you have 850 survey responses and 10% of them answered "yes" to the first question. Sample Size: You have been hired by a college foundation to conduct a survey of graduates. If you want to estimate the percentage of graduates who have made a donation to the college after graduation, how many graduates must you survey if you want 98% confidence that your percentage has a margin of error of five percentage points? If you want to estimate the mean amount of all charitable contributions made by graduates, how many graduates must you survey if you want 98% confidence that your sample mean is in error by no more than $50? (Based on results from a pilot study, assume that the standard deviation of donations by graduates is $337.) If you plan to obtain the estimates described in parts (a) and (b) with a single survey having several questions, how many graduates must be surveyed? Alcohol Consumption: In a Gallup poll, 1011 adults were asked if they consume alcoholic beverages and 64% of them said that they did. Construct a 90% confidence estimate of the proportion of all adults who consume alcoholic beverages. Can we conclude that the majority of adults consume alcoholic beverages? Wristwatch Accuracy: Students of the author collected data measuring the accuracy of wristwatches. The times (sec) below show the discrepancy between the real time and the time indicated on the wristwatch. Negative values correspond to watches that are running ahead of the actual time. The data satisfy a loose requirement of appearing to come from a normally distributed population. Construct a 95% confidence interval estimate of the mean discrepancy for the population of wristwatches. -85 325 20 305 -93 15 282 27 555 570 -241 36 White Blood Cell Counts: Data Set 1 in Appendix B lists the white blood cell counts (1000 cells/μL) of 40 randomly selected women. The mean of those 40 values is 7.15 and the standard deviation is 2.28. Construct a 90% confidence interval estimate of the mean white blood cell count for the population of women. Does the confidence interval also serve as an estimate of the mean white blood cell count of men? Car Crash Tests: Data Set 13 in Appendix B includes crash test measurements for small cars. The seven chest deceleration measurements have a mean of 42.7 g and a standard deviation of 5.6 g, where g is a force of gravity. Use the sample data to construct a 95% confidence interval estimate of the mean chest deceleration measurement for the population of all small cars. (Assume that the sample is a simple random sample and the measurements satisfy the loose requirement of being from a normally distributed population.) Write a brief statement that interprets the confidence interval. Car Crash Tests: Refer to the sample data described in Exercise 9 and construct a 95% confidence interval estimate of the population standard deviation.

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00:01 So we want to estimate how big our sample size needs to be if we want to find the percent who graduates who donate to their college at a 98 percent level.

00:16 The margin of error is an error that we want to stay below is 0 .05.

00:22 Since this is a percent, this is going to be a proportion.

00:30 So we're going to use the margin of error, which is a is the critical value times the square root of p hat 1 minus p hat over n.

00:55 Okay, we need to fill in these all with numbers.

01:01 So as we start over here, we're going to have 0 .05 is greater than or equal to.

01:08 And our critical value comes from our table.

01:12 So we're going to go to our table at 98%.

01:16 We are doing a proportion, but it doesn't really matter because whenever we're estimating a sample size, we want to use this very last line down here.

01:26 So this line that says 98 % has 2 .36.

01:33 So that's going to be our critical value.

01:37 2 .326.

01:40 Now, we don't have a p hat because that would be a sample proportion.

01:44 And they didn't give us any other proportions to use, so we're going to use the safest choice, which is 0 .5.

01:53 That's going to give us the largest variation in there.

01:57 And then over n.

01:59 N is what we're going to go through and solve for.

02:03 So, divide both sides by 2 .326, and then square both sides.

02:10 So i would have 0 .05 over 2 .326.

02:17 Squared is greater than or equal to 0 .5 over 0 .5 all divided by n.

02:26 Okay.

02:27 Now because this is on the bottom and this whole thing, even though it looks like a lot, is just going to be a number.

02:37 We can switch these guys.

02:40 So what i'm actually solving and what i can put in my calculator, that way i don't have any rounding issues, over 0 .5 over 2 .326 squared.

02:59 Okay, so i divided both sides by 2 .326 and got 0 .05 over 2 .326.

03:07 And then to get rid of the square root, i squared both sides.

03:10 That brought this out of the square root and squared this side.

03:14 And then if i multiply both sides by n and divide both sides by this number, i get to here.

03:20 And this is where i use my handy -dandy calculator to help me figure out what in the world is going on, because that is a mess of arithmetic to do in my head or on a piece of paper even.

03:37 So n has to be greater than or equal to 541 .0276.

03:44 So for part a, our n would need to be approximately 542.

03:53 People in order to make all of those things work now part b is going to be similar except for we're using the mean instead of the proportion okay and so that changes our margin of error it's still the critical value i'm going to abbreviate a little bit critical value times but this is going to be usually s sub x over the square root event okay, so they give us a few things here.

04:33 So the margin of error is 50.

04:36 So 50 has to be greater than or equal to the critical value.

04:42 We're still doing 98%.

04:45 So that means we're going to have the same critical value that we had in this part a.

04:50 So this is going to be 2 .326, just like it was up here.

04:57 And then we're going to multiply it.

04:59 They gave us a standard deviation, 337, and this would be like a previous years or decades or months or whatever standard deviation, and that would be over the square root of n.

05:14 Okay? and now we just need to do some algebra and get the end by itself.

05:19 So still the first step is the same.

05:23 We're going to divide by this guy, so let's do that.

05:39 Okay.

05:39 This is a number.

05:41 This is a number.

05:42 We're going to switch places with those or multiply both sides by this and divide both sides by this.

05:49 So i get square root of n is greater than or equal to 337 divided by 50 over 2 .326 and then square both sides.

06:06 And i'm going to put all of this into my calculator and figure out my n.

06:11 So this tells me that n would have to be bigger than or equal to 245 .7 .59.

06:39 So n should be approximately 246 people in order for us to get those results on our 98 % confidence number.

06:52 Okay, so then the next tool wants you to calculate some confidence intervals.

07:00 Okay, so this first one is about the accuracy of wrist watches, and the negative is that they're ahead of time, and the positive is that they're behind.

07:14 I believe that so i said.

07:17 So, 95 % confidence interval, and this is going to be a mean, not a proportion.

07:36 And so this is our formula here.

07:41 The statistic plus or minus the critical value times the standard error.

07:46 And for a mean, that means we're going to have the mean plus or minus the critical value.

07:54 I'm going to abbreviate to cv.

07:56 I hope that's okay...

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