Increasing the sample size of an opinion poll will reduce the

variability of the estimates made from the data collected in the poll.

bias of the estimates made from the data collected in the poll.

variability of opinions in the sample.

variability of opinions in the population.

These symbols represent the mean and standard deviation for which of the following distributions?

The Sampling Distribution of sample means

These symbols represent the mean and standard deviation for which of the following distributions?

The Sampling Distribution of sample means

These symbols represent the mean and standard deviation for which of the following distributions?

The Sampling Distribution of x-bar

What does this symbol represent?

standard deviation of the sample proportions

standard deviation of a parameter

standard deviation of the population

Sports Illustrated asked a random sample of 750 Division I college athletes, "Do you believe performance-enhancing drugs are a problem in college sports?" Suppose that 30% of all Division I athletes think that these drugs are a problem. Let p'hat' be the sample proportion who say that these drugs are a problem. Decreasing the sample size from 750 to 375 would have what affect on the standard deviation of the sampling distribution of p'hat'?

increase by a factor of 2 \sqrt{2} 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

decrease by a factor of 1 2 \frac{1}{\sqrt{2}} 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> 1

In a congressional district, 55% of the registered voters are Democrats. Which of the following is equivalent to the probability of getting less than 50% Democrats in a random sample of size 100?

P ( z < 0.50 − 0.55 0.55 ( 0.45 ) 100 ) P\left(z<\frac{0.50-0.55}{\sqrt{\frac{0.55\left(0.45\right)}{100}}}\right) P ⎝ ⎛ z < 1 0 0 0 . 5 5 ( 0 . 4 5 ) <path d="M1001,80H400000v40H1013.1s-83.4,268,-264.1,840c-180.7, 572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7, -7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744c-10,12,-21,25,-33,39s-32,39,-32,39 c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30c26.7,-32.7,52,-63,76,-91s52,-60,52,-60 s208,722,208,722c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5, -658.5c53.7,-170.3,84.5,-266.8,92.5,-289.5c4,-6.7,10,-10,18,-10z M1001 80H400000v40H1013z"> 0 . 5 0 − 0 . 5 5 ⎠ ⎞

P ( z < 0.50 − 0.55 100 ) P\left(z<\frac{0.50-0.55}{100}\right) P ( z < 1 0 0 0 . 5 0 − 0 . 5 5 )

P ( z < 0.55 − 0.50 0.55 ( 0.45 ) 100 ) P\left(z<\frac{0.55-0.50}{\sqrt{\frac{0.55\left(0.45\right)}{100}}}\right) P ⎝ ⎛ z < 1 0 0 0 . 5 5 ( 0 . 4 5 ) <path d="M1001,80H400000v40H1013.1s-83.4,268,-264.1,840c-180.7, 572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7, -7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744c-10,12,-21,25,-33,39s-32,39,-32,39 c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30c26.7,-32.7,52,-63,76,-91s52,-60,52,-60 s208,722,208,722c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5, -658.5c53.7,-170.3,84.5,-266.8,92.5,-289.5c4,-6.7,10,-10,18,-10z M1001 80H400000v40H1013z"> 0 . 5 5 − 0 . 5 0 ⎠ ⎞

P ( z < 0.50 − 0.55 100 ( 0.55 ) ( 0.45 ) ) P\left(z<\frac{0.50-0.55}{\sqrt{100\left(0.55\right)\left(0.45\right)}}\right) P ( z < 1 0 0 ( 0 . 5 5 ) ( 0 . 4 5 ) <path d="M263,681c0.7,0,18,39.7,52,119c34,79.3,68.167, 158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120c340,-704.7,510.7,-1060.3,512,-1067 c4.7,-7.3,11,-11,19,-11H40000v40H1012.3s-271.3,567,-271.3,567c-38.7,80.7,-84, 175,-136,283c-52,108,-89.167,185.3,-111.5,232c-22.3,46.7,-33.8,70.3,-34.5,71 c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1s-109,-253,-109,-253c-72.7,-168,-109.3, -252,-110,-252c-10.7,8,-22,16.7,-34,26c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26 s76,-59,76,-59s76,-60,76,-60z M1001 80H40000v40H1012z"> 0 . 5 0 − 0 . 5 5 )

P ( z < 0.55 − 0.50 100 ( 0.50 ) ( 0.50 ) ) P\left(z<\frac{0.55-0.50}{\sqrt{100\left(0.50\right)\left(0.50\right)}}\right) P ( z < 1 0 0 ( 0 . 5 0 ) ( 0 . 5 0 ) <path d="M263,681c0.7,0,18,39.7,52,119c34,79.3,68.167, 158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120c340,-704.7,510.7,-1060.3,512,-1067 c4.7,-7.3,11,-11,19,-11H40000v40H1012.3s-271.3,567,-271.3,567c-38.7,80.7,-84, 175,-136,283c-52,108,-89.167,185.3,-111.5,232c-22.3,46.7,-33.8,70.3,-34.5,71 c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1s-109,-253,-109,-253c-72.7,-168,-109.3, -252,-110,-252c-10.7,8,-22,16.7,-34,26c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26 s76,-59,76,-59s76,-60,76,-60z M1001 80H40000v40H1012z"> 0 . 5 5 − 0 . 5 0 )

Why is it important to check the 10% condition before calculating probabilities involving x ‾ \overline{x} x ?

To ensure that the observations in the sample are close to independent.

To reduce the variability of the sampling distribution of x ‾ \overline{x} x .

To ensure that the sampling distribution x ‾ \overline{x} x is approximately Normal.

To ensure that we can generalize the results to a larger population.

To ensure that x ‾ \overline{x} x will be an unbiased estimator of μ \mu μ .

A recent study was conducted to investigate the duration of time required to complete a certain manual dexterity task. The reported mean was 10.2 seconds with a standard deviation of 16.0 seconds. Suppose the reported values are the true mean and standard deviation for the population of subjects in the study. If a random sample of 144 subjects is selected from the population, what is the approximate probability that the mean of the sample will be more than 11.0 seconds?

Based on the values of the true mean and true standard deviation, it can be concluded that the population distribution is not normal and therefore the probability cannot be calculated.

Researchers are studying populations of two squirrels, the eastern gray and the western gray. For the eastern gray squirrel, about 22 percent of the population weighs over 0.5 kilogram (kg) . For the western gray squirrel, about 36 percent of the population weighs over 0.5 kg . A random sample of 60 squirrels will be selected from the population of eastern gray squirrels, and a random sample of 120 squirrels will be selected from the population of western gray squirrels. What is the mean of the sampling distribution of the difference in sample proportions (eastern minus western) ?

0.22 − 0.36 0.22-0.36 0 . 2 2 − 0 . 3 6

0.22 ( 0.78 ) 120 + 0.36 ( 0.64 ) 60 \sqrt{\frac{0.22\left(0.78\right)}{120}+\frac{0.36\left(0.64\right)}{60}} 1 2 0 0 . 2 2 ( 0 . 7 8 ) + 6 0 0 . 3 6 ( 0 . 6 4 ) <path d="M424,2478c-1.3,-0.7,-38.5,-172,-111.5,-514c-73, -342,-109.8,-513.3,-110.5,-514c0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5, 25c-5.7,9.3,-9.8,16,-12.5,20s-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13, -13s76,-122,76,-122s77,-121,77,-121s209,968,209,968c0,-2,84.7,-361.7,254,-1079 c169.3,-717.3,254.7,-1077.7,256,-1081c4,-6.7,10,-10,18,-10H400000v40H1014.6 s-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185c-2,6,-10,9,-24,9 c-8,0,-12,-0.7,-12,-2z M1001 80H400000v40H1014z">

0.22 ( 0.78 ) 60 + 0.36 ( 0.64 ) 120 \sqrt{\frac{0.22\left(0.78\right)}{60}+\frac{0.36\left(0.64\right)}{120}} 6 0 0 . 2 2 ( 0 . 7 8 ) + 1 2 0 0 . 3 6 ( 0 . 6 4 ) <path d="M424,2478c-1.3,-0.7,-38.5,-172,-111.5,-514c-73, -342,-109.8,-513.3,-110.5,-514c0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5, 25c-5.7,9.3,-9.8,16,-12.5,20s-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13, -13s76,-122,76,-122s77,-121,77,-121s209,968,209,968c0,-2,84.7,-361.7,254,-1079 c169.3,-717.3,254.7,-1077.7,256,-1081c4,-6.7,10,-10,18,-10H400000v40H1014.6 s-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185c-2,6,-10,9,-24,9 c-8,0,-12,-0.7,-12,-2z M1001 80H400000v40H1014z">

60 ( 0.22 ) − 120 ( 0.36 ) 60(0.22)-120(0.36) 6 0 ( 0 . 2 2 ) − 1 2 0 ( 0 . 3 6 )

0.36 − 0.22 0.36\ -\ 0.22 0 . 3 6 − 0 . 2 2

AP Stats Sampling Distributions

Authored by Tina Shutich

Mathematics

10th - 11th Grade

CCSS covered

Used 76+ times

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MULTIPLE CHOICE QUESTION

1 min • 1 pt

The Gallup Poll asked a random sample of 1785 adults whether they attended church during the past week. Let p-hat be the proportion of people in the sample who attended church. A newspaper report claims that 40% of all U.S. adults went to church last week. Suppose this claim is true.
What is the mean of the sampling distribution of p-hat?

0.4

0.44

0.56

0.6

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

28% of all Woodrow students believe Monday will be a snow day. You take a sample of 50 students and find that 15 of them believe Monday will be a snow day. What does 50 represent?

p-hat

standard deviation

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

28% of all Woodrow students believe Monday will be a snow day. You take a sample of 50 students and find that 15 of them believe Monday will be a snow day. What does 28 represent?

p-hat

standard deviation

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

The Gallup Poll asked a random sample of 1785 adults whether they attended church during the past week. Let p-hat be the proportion of people in the sample who attended church. A newspaper report claims that 40% of all U.S. adults went to church last week. Suppose this claim is true.
Calculate the standard deviation of the sampling distribution of sample proportions.

0.0116

0.0117

0.0118

0.0119

MULTIPLE CHOICE QUESTION

1 min • 1 pt

What happens to the shape of a sampling distribution of sample means as n increases?

It becomes narrower and bimodal.

It becomes narrower and more normal.

It becomes wider and skewed right.

It becomes wider and more normal.

MULTIPLE CHOICE QUESTION

1 min • 1 pt

The sampling distribution of a statistic is...

Using repeated samples and finding a probabilitity

The distribution of values taken by a statistic in all possible samples of the same sample size from the same population

The way that sample results differe from each other

the distribution of values in a sample of size n

A statistic is unbiased if...

the value of the statistic from any sample is NOT equal to the parameter

The value of the statistic from any sample IS equal to the parameter

The mean of the sampling distribution is NOT equal to the true value of the parameter

The mean of the sampling distribution IS equal to the value of the parameter.

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