Chapter 7: Sampling Distributions

Question 1

Define Parameter

Answer 1

a number that describes some characteristic of the population

Question 2

Define Statistic

Answer 2

a number that describes some characteristic of a sample. the value of a statistic can be computed directly from the sample data

Question 3

symbols for
population mean
sample mean
proportion population
proportion sample

Answer 3

mu
x bar
p
p hat

Question 4

Define Sampling Variability

Answer 4

the value of a statistic varies in repeated random sampling

Question 5

Define Sampling Distribution of a statistic

Answer 5

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

Question 6

Define Variability of a Statistic

Answer 6

described by the spread of its sampling distribution
This spread is determined primarily by the size of the random sample.
larger samples give smaller spread.

Question 7

Define Bias

Answer 7

a statistic used to estimate a parameter is an unbiased estimator if the mean of its sampling distribution is equal to the true value of the parameter being estimated
bias = our aim is off and we consistently miss the bull’s-eye in the same direction. out sample values do not center on the population value.

Question 8

Sampling Distribution of a Sample Proportion

Answer 8

mean: p hat = p
standard deviation: sigma (p hat) = sqrt ((p(1-p))/n) [measures the typical distance between a sample proportion and the population proportion p in samples of size n.
conditions:
1. 10% n<1/10N [allows the use of sigma (p)]
2. large count: np≥10 and n(1-p) ≥ 10 [allows to assume that sampling distribution is approximately normal]

Question 9

Describe the change in sampling distribution once sample size decreases

Answer 9

spread would increase since smaller sample size = more variability
standard deviation would increase
large count might fail –> not approximately normal

Question 10

finding the smallest sample size [Sampling Distribution of a Sample Proportion]

Answer 10

np≥10 and n(1-p)≥10

Question 11

Sampling Distribution of a Difference Between Two Proportions Mean

Answer 11

mean: p hat1-p hat2 = p1 - p2
standard deviation: sigma (p hat1 - p hat2) = sqrt (p1(1-p1)/n1 + p2(1-p2))/n2) [measures the typical distance between a sample proportion and the population proportion p in samples of size n.
conditions:
1. 10% n1<1/10N1 n2<1/10N2 [allows the use of sigma (p)]
2. large count: n1p1≥10 and n1(1-p1) ≥ 10 and n2p2≥10 and n2(1-p2) ≥ 10[allows to assume that sampling distribution is approximately normal]

Question 12

Central Limit Theorem

Answer 12

when n is sufficiently large, the sampling distribution of the sample mean [x bar] is approximately normal.

Question 13

Sampling Distribution of a Difference Between Two Means

Answer 13

mean: mu hat1-p hat2 = mu 1 - mu 2
standard deviation: sigma (x bar1 - x bar2) = sqrt (sigma 1 ^2 / n1 + sigma 2 ^2 / n2) [measures the typical distance between a sample proportion and the population proportion p in samples of size n.
conditions:
1. 10% n1<1/10N1 n2<1/10N2 [allows the use of sigma (p)]
2. large count:
- both populations are normal so the sampling distribution of x bar1 - x bar2 is normal
- both samples exceeds the size of 30 or one population is normal and the other has sample size greater than 30.