Sampling Distributions Chapter 8 Sampling Distributions

Distribution of the Sample Mean Section Distribution of the Sample Mean 8.1

Statistics such as are random variables since their value varies from sample to sample. As such, they have probability distributions associated with them. In this chapter we focus on the shape, center and spread of statistics such as .

The sampling distribution of a statistic is a probability distribution for all possible values of the statistic computed from a sample of size n. The sampling distribution of the sample mean is the probability distribution of all possible values of the random variable computed from a sample of size n from a population with mean μ and standard deviation σ.

Illustrating Sampling Distributions Step 1: Obtain a simple random sample of size n. Step 2: Compute the sample mean. Step 3: Assuming that we are sampling from a finite population, repeat Steps 1 and 2 until all distinct simple random samples of size n have been obtained.

Parallel Example 1: Sampling Distribution of the Sample Parallel Example 1: Sampling Distribution of the Sample Mean-Normal Population The weights of pennies minted after 1982 are approximately normally distributed with mean 2.46 grams and standard deviation 0.02 grams. Approximate the sampling distribution of the sample mean by obtaining 200 simple random samples of size n = 5 from this population.

The data on the following slide represent the sample means for the 200 simple random samples of size n = 5. For example, the first sample of n = 5 had the following data: 2.433 2.466 2.423 2.442 2.456 Note: = 2.444 for this sample

Sample Means for Samples of Size n = 5

The mean of the 200 sample means is 2 The mean of the 200 sample means is 2.46, the same as the mean of the population. The standard deviation of the sample means is 0.0086, which is smaller than the standard deviation of the population. The next slide shows the histogram of the sample means.

What role does n, the sample size, play in the standard deviation of the distribution of the sample mean? As the size of the sample increases, the standard deviation of the distribution of the sample mean decreases.

Parallel Example 2: The Impact of Sample Size on Sampling Variability Approximate the sampling distribution of the sample mean by obtaining 200 simple random samples of size n = 20 from the population of weights of pennies minted after 1982 (μ = 2.46 grams and σ = 0.02 grams)

The mean of the 200 sample means for n = 20 is still 2 The mean of the 200 sample means for n = 20 is still 2.46, but the standard deviation is now 0.0045 (0.0086 for n = 5). As expected, there is less variability in the distribution of the sample mean with n =20 than with n = 5.

The Mean and Standard Deviation of the Sampling Distribution of Suppose that a simple random sample of size n is drawn from a large population with mean μ and standard deviation σ. The sampling distribution of will have mean and standard deviation The standard deviation of the sampling distribution of is called the standard error of the mean and is denoted .

The Shape of the Sampling Distribution of If X is Normal If a random variable X is normally distributed, the sampling distribution of the sample mean is normally distributed.

Parallel Example 3: Describing the Distribution of the Sample Mean The weights of pennies minted after 1982 are approximately normally distributed with mean 2.46 grams and standard deviation 0.02 grams. What is the probability that in a simple random sample of 10 pennies minted after 1982, we obtain a sample mean of at least 2.465 grams?

Solution is normally distributed with =2.46 and P(Z > 0.79) = 1 – 0.7852 = 0.2148.

Parallel Example 3: Describing the Distribution of the Sample Mean The IQ, X, of humans is approximately normally distributed with mean μ = 100 and standard deviation σ = 15. Compute the probability that a simple random sample of size n = 10 results in a sample mean greater than 110. That is, compute P( >110).

Parallel Example 4: Sampling from a Population that is Not Normal The following table and histogram give the probability distribution for rolling a fair die: μ = 3.5, σ = 1.708 Note that the population distribution is NOT normal Face on Die Relative Frequency 1 0.1667 2 3 4 5 6

Estimate the sampling distribution of by obtaining 200 simple random samples of size n = 4 and calculating the sample mean for each of the 200 samples. Repeat for n = 10 and 30. Histograms of the sampling distribution of the sample mean for each sample size are given on the next slide.

Key Points from Example 4 The mean of the sampling distribution is equal to the mean of the parent population and the standard deviation of the sampling distribution of the sample mean is regardless of the sample size. The shape of the distribution of the sample mean becomes approximately normal as the sample size n increases, regardless of the shape of the underlying population.

The Central Limit Theorem Regardless of the shape of the underlying population, the sampling distribution of becomes approximately normal as the sample size, n, increases.

Solution: is approximately normally distributed Parallel Example 5: Using the Central Limit Theorem Suppose that the mean time for an oil change at a “10-minute oil change joint” is 11.4 minutes with a standard deviation of 3.2 minutes. If a random sample of n = 35 oil changes is selected, describe the sampling distribution of the sample mean. (b) If a random sample of n = 35 oil changes is selected, what is the probability the mean oil change time is less than 11 minutes? Solution: is approximately normally distributed with mean = 11.4 and std. dev. = . Solution: , P(Z < –0.74) = 0.23.

Parallel Example 5: Using the Central Limit Theorem The mean weight gain during pregnancy is 30 pounds, with a standard deviation of 12.9 pounds. Weight gain during pregnancy is skewed right. An obstetrician obtains a random sample of 35 low-income patients and determines their mean weight gain during pregnancy was 36.2 pounds. Does this result suggest anything unusual?

Why is the sampling distribution of approximately normal? Example Suppose that cars arrive at Burger King’s drive-through at the rate of 20 cars every hour between 12:00 noon to 1:00 pm. A random sample of 40 one-hour time periods between 12:00 noon to 1:00 pm is selected and has 22.2 as the mean number of cars arriving. Why is the sampling distribution of approximately normal? What is the mean and standard deviation of the sampling distribution of assuming that μ = 20 and σ = 20 ? What is the probability that a simple random sample of 40 one-hour time periods results in a mean of at least 22.1 cars? Is this unusual? What might we conclude?

Distribution of the Sample Proportion Section Distribution of the Sample Proportion 8.2

Point Estimate of a Population Proportion Suppose that a random sample of size n is obtained from a population in which each individual either does or does not have a certain characteristic. The sample proportion, denoted (read “p-hat”) is given by where x is the number of individuals in the sample with the specified characteristic. The sample proportion is a statistic that estimates the population proportion, p.

Parallel Example 1: Computing a Sample Proportion In a Quinnipiac University Poll conducted in May of 2008, 1,745 registered voters nationwide were asked whether they approved of the way George W. Bush is handling the economy. 349 responded “yes”. Obtain a point estimate for the proportion of registered voters who approve of the way George W. Bush is handling the economy. Solution:

Key Points Shape: As the size of the sample, n, increases, the shape of the sampling distribution of the sample proportion becomes approximately normal. Center: The mean of the sampling distribution of the sample proportion equals the population proportion, p. Spread: The standard deviation of the sampling distribution of the sample proportion decreases as the sample size, n, increases.

Sampling Distribution of For a simple random sample of size n with population proportion p: The shape of the sampling distribution of is approximately normal provided np(1 – p) ≥ 10. The mean of the sampling distribution of is The standard deviation of the sampling distribution of is

Sampling Distribution of The model on the previous slide requires that the sampled values are independent. When sampling from finite populations, this assumption is verified by checking that the sample size n is no more than 5% of the population size N (n ≤ 0.05N). Regardless of whether np(1 – p) ≥ 10 or not, the mean of the sampling distribution of is p, and the standard deviation is

Parallel Example 3: Describing the Sampling Distribution of Parallel Example 3: Describing the Sampling Distribution of the Sample Proportion According to a Time poll conducted in June of 2008, 42% of registered voters believed that gay and lesbian couples should be allowed to marry. Suppose that we obtain a simple random sample of 50 voters and determine which voters believe that gay and lesbian couples should be allowed to marry. Describe the sampling distribution of the sample proportion for registered voters who believe that gay and lesbian couples should be allowed to marry.

Solution The sample of n = 50 is smaller than 5% of the population size (all registered voters in the U.S.). Also, np(1 – p) = 50(0.42)(0.58) = 12.18 ≥ 10. The sampling distribution of the sample proportion is therefore approximately normal with mean=0.42 and standard deviation = (Note: this is very close to the standard deviation of 0.072 found using simulation in Example 2.)

Example Based on a study, 76% of Americans believe that the state of moral values in the United States is getting worse. Suppose we obtain a simple random sample of n=60 Americans and determine which believe that the state of the moral values in the United States is getting worse. Describe the sampling distribution of the sample proportion for Americans with this belief.

Parallel Example 4: Compute Probabilities of a Sample Proportion According to the Centers for Disease Control and Prevention, 18.8% of school-aged children, aged 6-11 years, were overweight in 2004. In a random sample of 90 school-aged children, aged 6- 11 years, what is the probability that at least 19% are overweight? Suppose a random sample of 90 school-aged children, aged 6-11 years, results in 24 overweight children. What might you conclude?

Solution n = 90 is less than 5% of the population size np(1 – p) = 90(.188)(1 – .188) ≈ 13.7 ≥ 10 is approximately normal with mean=0.188 and standard deviation = In a random sample of 90 school-aged children, aged 6-11 years, what is the probability that at least 19% are overweight? , P(Z > 0.05)=1 – 0.5199=0.4801

Solution is approximately normal with mean = 0.188 and standard deviation = 0.0412 Suppose a random sample of 90 school-aged children, aged 6-11 years, results in 24 overweight children. What might you conclude? , P(Z > 1.91) = 1 – 0.9719 = 0.0281. We would only expect to see about 3 samples in 100 resulting in a sample proportion of 0.2667 or more. This is an unusual sample if the true population proportion is 0.188.

Example According to the National Center for Health Statistics, 15% of all Americans have hearing trouble. In a random sample of 120 Americans, what is the probability at most 12% have hearing trouble? Suppose a random sample of 120 Americans who regularly listens to music using headphones results in 26 having hearing trouble. What might you conclude?

According to creditcard.com, 29% of adults do not own a credit card. Example According to creditcard.com, 29% of adults do not own a credit card. Suppose a random sample of 500 adults is asked, “Do you own a credit card?” Describe the sampling distribution of , the proportion of adults who do not own a credit card. What is the probability that in a random sample of 500 adults more than 30% do not own a credit card? What is the probability that in a random sample of 500 adults between 25% and 30% do not own a credit card? Would it be unusual for a random sample of 500 adults to result in 125 or fewer who do not own a credit card? Why?