Question

Parts being manufactured at a plant are supposed to weigh 40 grams. Suppose the distribution of weights has a Normal distribution with a mean of 40 grams and a standard deviation of 2 grams. Quality control inspectors randomly select 16 parts, weigh each, and then compute the sample average weight for the 16 parts. The sampling distribution of the sample mean is: a. cannot be determined because the sample size (16) is fairly small. b. approximately Normal with a mean of 40 grams and a standard deviation of 0.5 grams. c. meaningless in this problem because only one sample is being selected. d. exactly Normal with a mean of 40 grams and a standard deviation of 0.5 grams. Janice runs a business that uses lots of batteries. A sales representative has told her that the length of time his company's batteries last has a mean of 1,000 hours and a standard deviation of 25 hours. She records the lifetimes of 100 batteries and finds that the mean is 980 hours. What should she conclude about the sales representative's claim? a. It is not unreasonable; the z-statistic for her observed mean is -8.0. b. It is unreasonable; the true population mean is clearly much less than 1,000 hours. c. It is unreasonable; either the true population mean is less than 1,000 hours or the true population standard deviation is larger than 25 hours, or both. d. It is not unreasonable; the z-statistic for her observed mean is -0.8. The incomes in a certain large population of college teachers have a normal distribution with a mean of $75,000 and a standard deviation of $10,000. Four teachers are selected at random from this population to serve on a committee. What is the probability that their average salary is less than $65,000? [Hint: firstly consider the standardized z-statistic for the sample] a. essentially 0 b. 0.0228 c. 0.1587 d. 0.9772 In a survey of eating habits, 3,800 national adults were selected randomly and contacted by telephone. Respondents were asked, “Typically, how many times per week do you purchase food at a fast-food restaurant or convenience store?” On average, those surveyed purchased food at a fast-food restaurant or convenience store 2.4 times per week. Suppose the researchers that conducted the survey described above decide to conduct it again, but will now interview only 2,200 people. In the first survey, their sample average value was 2.4. Which of the following is true about the sample average value for the second survey? a. A different value for the sample average may result as sample averages (as estimates of the population value) are variable. b. By the law of large numbers, the smaller (second) survey will certainly produce a sample average farther from the true population average than the larger (first) survey. c. The average income computed from the sample of 1,000 people would be more accurate because smaller samples tend to be more homogeneous than larger samples. d. By the law of large numbers, it would again be 2.4. A statistic is said to be unbiased if a. the person computing it doesn't favor any particular outcome. b. the mean of its sampling distribution is equal to the true value of the parameter being estimated. c. both the person who calculated the statistic and the subjects whose responses make up the statistic were truthful. d. it is used for only honest purposes.

Name: parts being manufactured at a plant are supposed to weigh 40 grams suppose the distribution of weights has a normal distribution with mean 40 grams and a standard deviation 2 grams quality c 89765
Uploaded: 2022-08-31T05:10:53-08:00
Duration: 7 min 12 s
Channel: Ariana Nash
Description: parts being manufactured at a plant are supposed to weigh 40 grams suppose the distribution of weights has a normal distribution with mean 40 grams and a standard deviation 2 grams quality c 89765

The sampling distribution of the sample mean is:

a. cannot be determined because the sample size (16) is fairly small.
b. approximately Normal with a mean of 40 grams and a standard deviation of 0.5 grams.
c. meaningless in this problem because only one sample is being selected.
d. exactly Normal with a mean of 40 grams and a standard deviation of 0.5 grams.

Janice runs a business that uses lots of batteries. A sales representative has told her that the length of time his company's batteries last has a mean of 1,000 hours and a standard deviation of 25 hours. She records the lifetimes of 100 batteries and finds that the mean is 980 hours. What should she conclude about the sales representative's claim?

a. It is not unreasonable; the z-statistic for her observed mean is -8.0.
b. It is unreasonable; the true population mean is clearly much less than 1,000 hours.
c. It is unreasonable; either the true population mean is less than 1,000 hours or the true population standard deviation is larger than 25 hours, or both.
d. It is not unreasonable; the z-statistic for her observed mean is -0.8.

The incomes in a certain large population of college teachers have a normal distribution with a mean of $75,000 and a standard deviation of $10,000. Four teachers are selected at random from this population to serve on a committee. What is the probability that their average salary is less than $65,000?

[Hint: firstly consider the standardized z-statistic for the sample]

a. essentially 0
b. 0.0228
c. 0.1587
d. 0.9772

In a survey of eating habits, 3,800 national adults were selected randomly and contacted by telephone. Respondents were asked, “Typically, how many times per week do you purchase food at a fast-food restaurant or convenience store?” On average, those surveyed purchased food at a fast-food restaurant or convenience store 2.4 times per week.

Suppose the researchers that conducted the survey described above decide to conduct it again, but will now interview only 2,200 people. In the first survey, their sample average value was 2.4. Which of the following is true about the sample average value for the second survey?

a. A different value for the sample average may result as sample averages (as estimates of the population value) are variable.
b. By the law of large numbers, the smaller (second) survey will certainly produce a sample average farther from the true population average than the larger (first) survey.
c. The average income computed from the sample of 1,000 people would be more accurate because smaller samples tend to be more homogeneous than larger samples.
d. By the law of large numbers, it would again be 2.4.

A statistic is said to be unbiased if

a. the person computing it doesn't favor any particular outcome.
b. the mean of its sampling distribution is equal to the true value of the parameter being estimated.
c. both the person who calculated the statistic and the subjects whose responses make up the statistic were truthful.
d. it is used for only honest purposes.

Added by Santiago L.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

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08/31/2022

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Parts being manufactured at a plant are supposed to weigh 40 grams. Suppose the distribution of weights has a Normal distribution with a mean of 40 grams and a standard deviation of 2 grams. Quality control inspectors randomly select 16 parts, weigh each, and then compute the sample average weight for the 16 parts. The sampling distribution of the sample mean is: a. cannot be determined because the sample size (16) is fairly small. b. approximately Normal with a mean of 40 grams and a standard deviation of 0.5 grams. c. meaningless in this problem because only one sample is being selected. d. exactly Normal with a mean of 40 grams and a standard deviation of 0.5 grams. Janice runs a business that uses lots of batteries. A sales representative has told her that the length of time his company's batteries last has a mean of 1,000 hours and a standard deviation of 25 hours. She records the lifetimes of 100 batteries and finds that the mean is 980 hours. What should she conclude about the sales representative's claim? a. It is not unreasonable; the z-statistic for her observed mean is -8.0. b. It is unreasonable; the true population mean is clearly much less than 1,000 hours. c. It is unreasonable; either the true population mean is less than 1,000 hours or the true population standard deviation is larger than 25 hours, or both. d. It is not unreasonable; the z-statistic for her observed mean is -0.8. The incomes in a certain large population of college teachers have a normal distribution with a mean of $75,000 and a standard deviation of $10,000. Four teachers are selected at random from this population to serve on a committee. What is the probability that their average salary is less than $65,000? [Hint: firstly consider the standardized z-statistic for the sample] a. essentially 0 b. 0.0228 c. 0.1587 d. 0.9772 In a survey of eating habits, 3,800 national adults were selected randomly and contacted by telephone. Respondents were asked, “Typically, how many times per week do you purchase food at a fast-food restaurant or convenience store?” On average, those surveyed purchased food at a fast-food restaurant or convenience store 2.4 times per week. Suppose the researchers that conducted the survey described above decide to conduct it again, but will now interview only 2,200 people. In the first survey, their sample average value was 2.4. Which of the following is true about the sample average value for the second survey? a. A different value for the sample average may result as sample averages (as estimates of the population value) are variable. b. By the law of large numbers, the smaller (second) survey will certainly produce a sample average farther from the true population average than the larger (first) survey. c. The average income computed from the sample of 1,000 people would be more accurate because smaller samples tend to be more homogeneous than larger samples. d. By the law of large numbers, it would again be 2.4. A statistic is said to be unbiased if a. the person computing it doesn't favor any particular outcome. b. the mean of its sampling distribution is equal to the true value of the parameter being estimated. c. both the person who calculated the statistic and the subjects whose responses make up the statistic were truthful. d. it is used for only honest purposes.

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Transcript

00:01 Okay, so we have a few different parts here.

00:03 One thing we know that the distribution of weights has a normal distribution with a mean of 40 and a certain deviation of 2.

00:11 And we're supposed to describe the sampling distribution when we have an equal 16.

00:15 So since this is a normal distribution, we know automatically that the sampling distribution will also be normal.

00:23 So the sampling distribution will also be exactly normal.

00:28 It'll have the same mean as the population.

00:32 And it's serend deviation of the sampling distribution is going to be sigma over the square root of n, which is 2 over 4, which is 100.

00:44 That's always going to be the case where if you have a normal population, then no matter what the sample sizes, you know that the sampling distribution is going to be normal as well.

00:52 Okay, next thing is we have a mean that we think is 1 ,000 with a certain deviation of 25, and the janus, she records the lack of times of 100 batteries and finds that the mean of those 100 batteries is 980 and asks what to do about the claim.

01:12 So we need to just find a z really quickly.

01:14 So that's going to be z equals x bar minus mu over sigma over the square of n and see how unusual the sample that is.

01:23 So we'll take 980 minus 1 ,000 divided by 25 over the square of 100.

01:31 So that's going to be negative 20 over 25 divided by 10, which is 2 .5.

01:39 If we take negative 20 divided by 2 .5, we get negative 8.

01:48 Okay, so we have a few different parts here.

01:51 One thing we know that the distribution of weights has a normal distribution with a mean of 40 and a certain deviation of 2.

01:59 And we're supposed to describe the sampling.

02:00 So that's a very large test statistic.

02:05 So i'm going to go with c.

02:07 The claim is unreasonable.

02:10 Either the true mean is less than 1 ,000, which is what this says here, right? or it could also be that the serenade deviation is larger than 25 hours.

02:18 Because if the serend deviation was much larger, say like 100, then we would have a test statistic of negative 20 over 10, right, which would be much, much smaller.

02:30 So we're going to say c for that one.

02:34 Okay.

02:35 Next we have a normal distribution with a mean of 75 ,000 and certain deviation of 10 ,000.

02:42 And four teachers are selected at random.

02:47 We want the probability that their average seller is less than $65 ,000.

02:54 Okay, so we're going to, once again, take a z test statistic...

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