7.1: What is a Sampling Distribution?!?!teachers.dadeschools.net/sdaniel/AP Stats Chapter 7...

7.1: What is a Sampling Distribution?!?!

Section 7.1 What Is a Sampling Distribution?

After this section, you should be able to…

✓DISTINGUISH between a parameter and a statistic

✓DEFINE sampling distribution

✓DISTINGUISH between population distribution, sampling distribution, and the distribution of sample data

✓DETERMINE whether a statistic is an unbiased estimator of a population parameter

✓DESCRIBE the relationship between sample size and the variability of an estimator

The process of statistical inference involves using information from a sample to draw conclusions about a wider population. Different random samples yield different statistics.

We need to be able to describe the sampling distribution of possible statistic values in order to perform statistical inference. We can think of a statistic as a random variable because it takes numerical values that describe the outcomes of the random sampling process.

Population

Sample Collect data from a

representative Sample...

Make an Inference

about the Population.

Parameters and Statistics

A parameter is a number that describes some characteristic of the population. In statistical practice, the value of a parameter is usually not known because we cannot examine the entire population.

A statistic is a number that describes some characteristic of a sample. The value of a statistic can be computed directly from the sample data. We use a statistic to estimate an unknown parameter.

Symbols: Parameters and Statistics

Proportions Means StandardDeviation

Statistic s

Parameter p µ

Parameter v. Statistic

Identify the population, the parameter (of interest), the sample, and the statistic in each of the following settings.

A pediatrician wants to know the 75th percentile for the distribution of heights of 10-

year-old boys so she takes a sample of 50 patients and calculates Q3 = 56 inches.


Population: all 10-year-old boys

Parameter: 75th percentile, or Q3

Sample: 50 10-year-old boys included in the sample

Statistic: Q3 = 56 inches.


Identify the population, the parameter, the sample, and the statistic in each of the following settings.

A Pew Research Center poll asked 1100 12 to 17-year-olds in the United States if they

have a cell phone. Of the respondents, 71% said yes.


Population: All 12-17 year olds in the US

Parameter: Proportion with cell phones

Sample: 1100 12-17 year olds with cell phones

Statistic: Ƹ𝑝 = 0.71

Sampling Distribution

The sampling distribution of a statistic is the distribution of values taken by the statistic in ALL possible samples of the same size from the same population.

In practice, it’s difficult (usually impossible) to take all possible samples of size n to obtain the actual sampling distribution of a statistic. Instead, we can use simulation to imitate the process of taking many, many samples.

One of the uses of probability theory in statistics is to obtain sampling distributions without simulation. We’ll get to the theory later.

Population Distributions vs. Sampling Distributions

There are actually three distinct distributions involved when we sample repeatedly and measure a variable of interest.

1) The population distribution gives the values of the variable for all the individuals in the population.

2) The distribution of sample data shows the values of the variable for all the individuals in the sample.

3) The sampling distribution shows the statistic values from all the possible samples of the same size from the population.

Hours of Sleep Activity

1. Write your name and the number of hours of sleep (e.g., 7 hours, 8.5 hours) on the paper provided.

2. Select a SRS of 5 cards. Each person will do this. (Ignore sampling independence concerns).

3. Using your values calculate the sample IQR of sleep hours and the sample maximum of sleep hours. Then, plot your values on the board.

4. Based on these values and the approximate sampling distributions, do either of these statistics appear to be unbiased estimators?

Bias & Variability

Bias means that our aim is off and we consistently miss the bull’s-eye in the same direction. Our sample values do not center on the population value.

High variability means that repeated shots are widely scattered on the target. Repeated samples do not give very similar results.

Describing Sampling Distributions: Center

A statistic used to estimate a parameter is an unbiased estimator (most accurate) if the mean of its sampling distribution is equal to the true value of the parameter being estimated.

Describing Sampling Distributions: Spread

The variability of a statistic is 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. The spread of the sampling distribution does not depend on the size of the population, as long as the population is at least 10 times larger than the sample.

n=100 n=1000

Describing Sampling Distributions: Shape

Sampling distributions can take on many shapes. The same statistic can have sampling distributions with different shapes depending on the population distribution and the sample size.

Sampling distributions for different statistics used to

estimate the number of tanks in German during World War 2.

The blue line represents the true number of tanks.

A. Which of these statistics appear to be biased estimators?

B. Of the unbiased estimators, which is the best? Explain.

7.2: Sample Proportions

Section 7.2Sample Proportions


✓FIND the mean and standard deviation of the sampling distribution of a sample proportion

✓DETERMINE whether or not it is appropriate to use the Normal approximation to calculate probabilities involving the sample proportion

✓CALCULATE probabilities involving the sample proportion

✓EVALUATE a claim about a population proportion using the sampling distribution of the sample proportion

http://www.rossmanchance.com/applets/Reeses/ReesesPieces.html

http://www.rossmanchance.com/applets/Reeses/ReesesPieces.html

The Sampling Distribution of

What do you notice about the shape, center, and spread of each?

n =100 n =400

Sample Proportion Formulas

n

ppp

)1(ˆ

The sample size MUST be less than 10% of the total population.

Normal Approximation & Sample Proportions

As the sample size increase, sample proportion approach the normal distribution; therefore, we can use Normal calculations.

Before using Normal calculation, check Normal conditions:

– (sample size)(proportion) must be greater than 10.

– (sample size)(1 – proportion) must be greater than 10.

– Both must be greater than 10


In the game of Scrabble, each player starts by drawing 7 tiles from a bag of 100 tiles. There are 42 vowels, 56 constants and 2 blank tiles. Cait choses an SRS of 7 tiles. Let be the proportion of vowels in her sample.

a) Is the 10% condition met? Justify your answer.

b) Is the Normal condition met? Justify your answer.


(a) Yes. Seven tiles is less than 10% of the population of 100 tiles.

(b) No. Since the total sample size was 7, both np and n(1-p) must be less than 10. The Normal condition is not satisfied.

A polling organization asks an SRS of 1500 first-year college students how far away their home is. Suppose that 35% of

all first-year students actually attend college within 50 miles of home.

What is the probability that the random sample of 1500 students will give a result within 2 percentage points of

this true value?


We have an SRS of size n = 1500 drawn from a population in which the proportion p = 0.35 attend college within 50 miles of home.

Conditions:Independence: It is reasonable to assume that there are more than 15,000 college freshmen and therefore the sample represents less than 10% of the population.

Normality: Additionally, np = 1500(0.35) = 525 and n(1 –p) = 1500(0.65)=975 are both greater than 10, so it is reasonable to assume normality.

ˆ p 0.35

ˆ p (0.35)(0.65)

1500 0.0123

Normalcdf (0.33, 0.37, 0.35, 0.0123) = 0.896054

CONCLUDE: There is an 89.61% chance that the sample will yield results within 2 percentage points of the true value.

The Harvard College Alcohol Study finds that 67% of college students support efforts to “crack down on underage drinking.” The study took a random sample of almost 15,000 students, so

the population proportion who support a crackdown is close to p= 0.67. The administration of a local college surveys an SRS of

100 students and finds that 62 support a crackdown on underage drinking. Suppose that the proportion of all students attending this college who support a crackdown is 67%, the same as the

national proportion.

What is the probability that the proportion in an SRS of 100 students is as small as or smaller than the result of

the administration’s sample?

Conditions:Independence: It is reasonable to assume that there are more than 1000 college freshmen and therefore the sample represents less than 10% of the population.

Normality: Additionally, np = 100(0.67) = 67and n(1 – p) = 100(0.33)= 33 are both greater than 10, so it is reasonable to assume normality.

67.0ˆ p 04702.0100

)33.0)(67.0(ˆ p

Normalcdf (0, 0.62, 0.67, 0.04702) = 0.143805

Be sure to include labels!

CONCLUDE: There is an 14.38% chance that the sample will yield results at or below 62% given that the true population proportions is 67%

FYI: Derivation of Formulas In Chapter 6, we learned that the mean and standard deviation of a binomial random variable X are

X np

X np(1 p)

ˆ p 1

nnp(1 p)

np(1 p)

n2

p(1 p)

n

ˆ p 1

n(np) p

As sample size increases, the spread decreases.

Since ˆ p X /n (1/n) X, we are just multiplying the random variable X

by a constant (1/n) to get the random variable ˆ p . Therefore,

ˆ p is an unbiased estimator or p

7.3: Sample Means

Section 7.3Sample Means


✓FIND the mean and standard deviation of the sampling distribution of a sample mean

✓CALCULATE probabilities involving a sample mean when the population distribution is Normal

✓EXPLAIN how the shape of the sampling distribution of sample means is related to the shape of the population distribution

✓APPLY the central limit theorem to help find probabilities involving a sample mean

Sample MeansConsider the mean household earnings for samples of size 100. Compare the population distribution on the left with the sampling distribution on the right. What do you notice about the shape, center, and spread of each?

Theory: Sample Means

Sample Means Formulas

Notes: The sample size must be less than 10% of the population to satisfy the independence condition. The mean and standard deviation of the sample mean are true no matter the same of the population distribution.

x

REVIEW: Young Women’s Heights

The height of young women follows a Normal distribution with mean µ = 64.5 inches and standard deviation σ = 2.5

inches.

Find the probability that a randomly selected young woman is taller than 66.5 inches.

REVIEW: Young Women’s Heights

STATE: Let X = the height of a randomly selected young woman. X is N(64.5, 2.5).

PLAN: Since the “sample” in this case is only one person, the sample size is clearly smaller than the 10% of the population.

DO:

OR Normalcdf (66.5, 10000, 64.5, 2.5) = 0.2118

P(X 66.5) P(Z 0.80) 10.7881 0.2119

z 66.5 64.5

2.5 0.80

CONCLUDE: The probability of choosing a young woman at random whose height exceeds 66.5 inches is about 0.21.

Example: Young Women’s Heights

The height of young women follows a Normal distribution with mean µ = 64.5 inches and standard deviation σ = 2.5

inches.

Find the probability that the mean height of an SRS of 10 young women exceeds 66.5 inches.

Example: Young Women’s Heights

z 66.5 64.5

0.79 2.53

P(x 66.5) P(Z 2.53)

10.9943 0.0057

CONCLUDE: There is a 0.57% percent chance of getting a sample of 10 women with a mean height of 66.5 It is very unlikely (less than a 1% chance) that we would choose an SRS of 10 young women whose average height exceeds 66.5 inches.

OR normalcdf(66.5, 10000, 64.5, 0.7905) = 0.0057

Sample Distributions & Normality

If the population is Normal, then the sample distribution is Normal. No further checks are need!

If the population is NOT Normal, then….

If the sample is large enough, the distribution of sample means is “approximately” Normal, no matter what shape the population distribution has, as long as the population has a finite standard deviation.

Sample Distributions & Normality:

Sample Distributions & Normality:

HOW LARGE IS LARGE ENOUGH?

If the Population shape is….

Minimum Sample Size to assume Normal

Normal 0

Slightly Skewed 15

Heavily Skewed 30

Unknown 30

Example: Servicing Air Conditioners

Based on many service records from the past year, the time (in hours) that a technician requires to complete preventative maintenance on an air conditioner follows the distribution that is strongly right-skewed, and whose most likely outcomes are close to 0. The mean time is µ = 1 hour and the standard deviation is σ = 1

Your company will service an SRS of 70 air conditioners. You have budgeted 1.1 hours per unit. Will this be

enough time? What is the chance that the technician will not finish within the allotted time (1.1 hours) ?


x 1

Conditions:Independence: It is reasonable to assume that the company has serviced more than 700 unit, therefore the 70 units in the sample represent less than 10% of the population.

Normal: Even though the population has a strong right skew, a sample size of 70 is large enough to assume normality.

x

n

1

70 0.12

𝜎𝜇 S


DO:

z 1.11

0.12 0.83

P(x 1.1) P(Z 0.83)

10.7967 0.2033

CONCLUDE: If you budget 1.1 hours per unit, there is a 20.13% chance the technicians will not complete the work within the budgeted time.

OR Normalcdf (1.1000001, 10000, 1, 0.1195) = 0.2013

7.1: What is a Sampling Distribution?!?!teachers.dadeschools.net/sdaniel/AP Stats Chapter 7...

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