Section 9.3

Sample Means

Sample Means

The mean and standard deviation of a

population are parameters we write these

as µ and σ

The mean and standard deviation calculated

from sample data are statistics we write

these as and s x

Sampling Distribution of a sample mean

Draw an SRS of size n from a population that has the normal distribution with mean µ and standard deviation σ. Then the sample

mean has the normal distribution N(µ, )

xn

THE CENTRAL LIMIT THEOREM

Central Limit Theorem states that for large n

the sampling distribution of is approximately

normal for any population with finite standard

deviation σ. The mean and standard deviation of

the normal distribution are the mean µ and

standard deviation .

Like before (with proportions), we want our

population to be at least ten times as big as the

sample to use this recipe for standard deviation

x

n

Law of Large Numbers

Law of Large Numbers states that the

actually observed mean outcome of a

large number of observations must

approach the mean µ of the population as

the number of observations increases.

x

Let’s Try Some Multiple Choice NUMBER ONE

When a sample is randomly selected from a population, what is most likely to occur?

– A. The sample distribution will be identical to the population distribution.

– B. The sample distribution will be quite similar to the population distribution at first, but will become less similar as the sample gets larger.

– C. The sample distribution could be quite different from the population distribution at first, but will become more similar as the sample gets larger.

– D. The sample distribution will not be at all similar to the population distribution.

AND THE ANSWER IS. . .

Correct Answer: C — The sample distribution could be quite different from the population distribution at first, but will become more similar as the sample gets larger. Explanation: Small samples are often quite different from the populations they are from, but as the sample becomes larger the sample tends to resemble the population distribution more and more. This is true in terms of the overall shape of the distribution, as well as the sample mean compared with the population mean.

Let’s Try Some Multiple Choice NUMBER TWO

When samples are randomly selected from a population, what is most likely to occur?

– A. The sampling distribution will be identical to the population distribution.

– B. The sampling distribution will be quite similar to the population distribution at first, but will become more normal as the samples gets larger.

– C. The sampling distribution could be quite different from the population distribution at first, but will become more similar as the samples gets larger.

– D. The sampling distribution will not be at all similar to the population distribution.

AND THE ANSWER IS. . .

Correct Answer: B — The sampling distribution will

be quite similar to the population distribution at first,

but will become more normal as the samples get

larger.

Explanation: Don’t forget, the sampling distribution is

different than a sample distribution. The Central

Limit Theorem reminds us that the sampling

distribution will be approximately normal for large

sample sizes regardless of the shape of the

population distribution. This is what makes the

Central Limit Theorem so important!

NUMBER THREE Let’s Try Some Multiple Choice Suppose you randomly selected a group of American

adults and asked them how many hours of TV they

watched this past Saturday, and the average of their answers was 4 hours. Which of the following would be most likely to be the distribution of their answers?

AND THE ANSWER IS. . .

Correct Answer: D — Distribution D Explanation: Distribution D shows most responses between 0 and 5 hours, with a short tail on the right. Distribution A is not likely because it is extremely unlikely that the number of people who watch 8 hours of TV in a day is the same as the number of people who watch 1 hour of TV. Distribution B is too tightly bunched; surely not everyone you sample would have watched 3, 4, or 5 hours or TV. Finally, distribution C is impossible because it appears that someone in the sample responded that he or she watched 25 hours of TV on Saturday.

Which of the sample distributions on the next

slide is most likely to be from this population

distribution?

NUMBER FOUR Let’s Try Some Multiple Choice

Choices for Number Four

AND THE ANSWER IS. . .

Correct Answer: D — Sample D Explanation: Only sample D resembles the given population sample. Even though sample A has exactly the same mean as the population, the values in sample A are too spread out (ranging from 0 to 20). Sample B is has only the values 9, 10, and 11 in it, and that is highly unlikely to come from the given population. Sample C appears to have far more values near 5 and 15, and fewer near 10, than the population.

Which of the following population distributions is more likely to be well

represented by a very small sample?

– A. The top (red) distribution would be better represented by a small sample because it is more spread out.

– B. The bottom (blue) distribution would be better represented by a small sample because it is less spread out.

– C. There should be no difference in how well the two populations are approximated by a small sample as long as the samples are randomly selected.

– D. The two cannot be compared because their means are different.

NUMBER FIVE Let’s Try Some Multiple Choice

AND THE ANSWER IS. . .

Correct Answer: B — The bottom (blue) distribution would be better represented by a small sample because it is less spread out. Explanation: The bottom (blue) distribution is fairly tightly clustered, so even a small sample should produce a mean that is fairly near the true population mean. The top (red) distribution is much more spread out, so there is a greater chance that a small sample could select values far from the center of the distribution. This would cause the sample mean to be relatively far from the population mean.

Free Response #1: Hypokalemia

Judy’s doctor is concerned that she may suffer from hypokalemia (low potassium in the blood). There is variation both in the actual potassium level and in the blood

test that measures the level. Judy’s measured potassium level varies according to the normal distribution with μ=3.8 and σ =0.2. A patient is classified as hypokalemic if the potassium level is below 3.5.

a) If a single potassium measurement is made, what is the probability that Judy is diagnosed as hypokalemic?

b) If measurements are made instead on 4 separate days and the mean result is compared with the criterion 3.5, what is the probability Judy is diagnosed as hypokalemic?

normalcdf(-E99, 3.5, 3.8, 0.2) ≈ 0.0668

normalcdf(-E99, 3.5, 3.8, 0.2/√(4)) ≈ 0.0013

Free Response #2: I-pod Songs David’s i-pod has about 10,000 songs. The distribution of

the play time for these songs is heavily skewed to the right with a mean of 225 seconds and a standard deviation of 60 seconds.

We choose a SRS of 10 songs from this population and calculate the mean play time of these songs.

a) What are the mean and the standard deviation of the sampling distribution of ?

b) Can you safely calculate the probability that the mean play time is more than 4 minutes (240 seconds) for an SRS of 10 songs? Why or why not?

Since the population distribution of play times on David’s i-pod is heavily

skewed to the right, we cannot use Normal approximation to calculate

probability with a sample size of 10 songs.

x

225seconds

6018.974seconds

10

x

xx

n

Free Response #2: I-pod Songs David’s i-pod has about 10,000 songs. The distribution of

the play time for these songs is heavily skewed to the right with a mean of 225 seconds and a standard deviation of 60 seconds.

c) Suppose that we take an SRS of 36 songs instead. Could we calculate the probability that the mean play time is more than 240 seconds? Explain why or why not.

According to the Central Limit Theorem, we can use Normal approximation

to calculate probability when the sample size is large enough. Since the

sample size is now 36, we have enough songs in the sample for the CLT to

apply.

Free Response #2: I-pod Songs d) . David’s i-pod has about 10,000 songs. The distribution of the play time for these songs is heavily skewed to the right with a mean of 225 seconds and a standard deviation of 60 seconds. Using a sample size of 36, calculate the probability that the average play time is greater than 4 minutes. Show ALL of the steps.