Sampling Distributions – Sample Means & Sample Proportions Chapter 18 Sampling Distributions – Sample Means & Sample Proportions

Parameter A number that describes the population Symbols we will use for parameters include m - mean s – standard deviation p/ p – proportion (p) a – y-intercept of LSRL b – slope of LSRL

Statistic A number that can be computed from sample data without making use of any unknown parameter Symbols we will use for statistics include x – mean s – standard deviation p – proportion a – y-intercept of LSRL b – slope of LSRL

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.

What is the mean and standard deviation of this population? Consider the population of 5 fish in my pond – the length of fish (in inches): 2, 7, 10, 11, 14 What is the mean and standard deviation of this population? mx = 8.8 sx = 4.0694

Let’s take samples of size 2 (n = 2) from this population: How many samples of size 2 are possible? 5C2 = 10 mx = 8.8 Find all 10 of these samples and record the sample means. What is the mean and standard deviation of the sample means? sx = 2.4919

Repeat this procedure with sample size n = 3 How many samples of size 3 are possible? 5C3 = 10 mx = 8.8 What is the mean and standard deviation of the sample means? Find all of these samples and record the sample means. sx = 1.66132

What do you notice? mx = m as n sx The mean of the sampling distribution EQUALS the mean of the population. As the sample size increases, the standard deviation of the sampling distribution decreases. mx = m as n sx

Remember the Jelly Blubbers? A statistic used to estimate a parameter is unbiased if the mean of its sampling distribution is equal to the true value of the parameter being estimated. Remember the Jelly Blubbers? The judgmental samples were centered too high & were bias, while the randomly selected samples were centered over the true mean

General Properties mx = m s sx = n Rule 1: Rule 2: This rule is approximately correct as long as no more than (10%) of the population is included in the sample mx = m s n sx =

General Properties Rule 3: When the population distribution is normal, the sampling distribution of x is also normal for any sample size n.

Activity – drawing samples

How large is “sufficiently large” anyway? General Properties Rule 4: Central Limit Theorem When n is sufficiently large, the sampling distribution of x is well approximated by a normal curve, even when the population distribution is not itself normal. How large is “sufficiently large” anyway? CLT can safely be applied if n exceeds 30.

EX) The army reports that the distribution of head circumference among soldiers is approximately normal with mean 22.8 inches and standard deviation of 1.1 inches. a) What is the probability that a randomly selected soldier’s head will have a circumference that is greater than 23.5 inches? P(X > 23.5) = .2623

What normal curve are you now working with? b) What is the probability that a random sample of five soldiers will have an average head circumference that is greater than 23.5 inches? Do you expect the probability to be more or less than the answer to part (a)? Explain What normal curve are you now working with? P(X > 23.5) = .0774

Suppose a team of biologists has been studying the Pinedale children’s fishing pond. Let x represent the length of a single trout taken at random from the pond. This group of biologists has determined that the length has a normal distribution with mean of 10.2 inches and standard deviation of 1.4 inches. What is the probability that a single trout taken at random from the pond is between 8 and 12 inches long? P(8 < X < 12) = .8427

P(8< x <12) = .9978 x = 11.43 inches What is the probability that the mean length of five trout taken at random is between 8 and 12 inches long? What sample mean would be at the 95th percentile? (Assume n = 5) Do you expect the probability to be more or less than the answer to part (a)? Explain P(8< x <12) = .9978 x = 11.43 inches

A soft-drink bottler claims that, on average, cans contain 12 oz of soda. Let x denote the actual volume of soda in a randomly selected can. Suppose that x is normally distributed with s = .16 oz. Sixteen cans are randomly selected and a mean of 12.1 oz is calculated. What is the probability that the mean of 16 cans will exceed 12.1 oz? P(x >12.1) = .0062

A hot dog manufacturer asserts that one of its brands of hot dogs has a average fat content of 18 grams per hot dog with standard deviation of 1 gram. Consumers of this brand would probably not be disturbed if the mean was less than 18 grams, but would be unhappy if it exceeded 18 grams. An independent testing organization is asked to analyze a random sample of 36 hot dogs. Suppose the resulting sample mean is 18.4 grams. What is the probability that the sample mean is greater than 18.4 grams? P(x >12.1) = .0082

Does this result indicate that the manufacturer’s claim is incorrect? Yes, not likely to happen by chance alone.

Modeling the Distribution of Sample Proportions Rather than showing real repeated samples, imagine what would happen if we were to actually draw many samples. Now imagine what would happen if we looked at the sample proportions for these samples. What would the histogram of all the sample proportions look like?

Modeling the Distribution of Sample Proportions (cont.) We would expect the histogram of the sample proportions to center at the true proportion, p, in the population. As far as the shape of the histogram goes, we can simulate a bunch of random samples that we didn’t really draw.

Modeling the Distribution of Sample Proportions (cont.) It turns out that the histogram is unimodal, symmetric, and centered at p. More specifically, it’s an amazing and fortunate fact that a Normal model is just the right one for the histogram of sample proportions. To use a Normal model, we need to specify its mean and standard deviation. The mean of this particular Normal is at p.

The Sampling Distribution Model for a Proportion (cont.) Provided that the sampled values are independent and the sample size is large enough, the sampling distribution of is modeled by a Normal model with Mean: Standard deviation:

Assumptions and Conditions Most models are useful only when specific assumptions are true. There are two assumptions in the case of the model for the distribution of sample proportions: The sampled values must be independent of each other. The sample size, n, must be large enough.

Assumptions and Conditions (cont.) 10% condition: If sampling has not been made with replacement, then the sample size, n, must be no larger than 10% of the population (population is at least 10 times as large as the sample) Success/failure condition: The sample size has to be big enough so that both and are greater than or equal to 10.

EXAMPLE You ask an SRS of 1500 first year college students whether they applied to any other college. There are over 1.7 million first year college students. 35% of all first year students applied to other colleges. What is the probability that your sample will give a result within 2 percentage points of this true value?

EXAMPLE CONTINUED Step 1: Calculate the mean and standard deviation Step 2: Standardize the scores

EXAMPLE CONTINUED Step 1: Calculate the mean and standard deviation Step 2: Standardize the scores

EXAMPLE CONTINUED Step 3: Find the So almost 90% of all samples will give a result within 2 percentage points of the true value of the population

What have we learned? We know that no sample fully and exactly describes the population; sample proportions and means will vary from sample to sample. That’s sampling error (or, better, sampling variability). We know it will always be present – indeed, the world would be a boring place if variability didn’t exist. You might think that sampling variability would prevent us from learning anything reliable about a population by looking at a sample, but that’s just not so. The fortunate fact is that sampling variability is not just unavoidable – it’s predictable!

What have we learned? (cont.) We’ve learned to describe the behavior of sample proportions when our sample is random and large enough to expect at least 10 successes and failures. We’ve also learned to describe the behavior of sample means (thanks to the CLT!) when our sample is random (and larger if our data come from a population that’s not roughly unimodal and symmetric).

What Can Go Wrong? (cont.) Beware of observations that are not independent. The CLT depends crucially on the assumption of independence. You can’t check this with your data—you have to think about how the data were gathered. Watch out for small samples from skewed populations. The more skewed the distribution, the larger the sample size we need for the CLT to work.

ASSIGNMENT p. 428 #27, 28, 33, 38 (online) #21, 22, 23, 26 (book)