Proportions – Sampling Distributions & Inference



Confidence Intervals Notes

What happens to your confidence as the interval gets smaller?

Point Estimate:

Confidence Interval:

Margin of Error:

Confidence Level:

Critical Value:

Confidence Interval for a Proportion:

What does it mean to be 95% confident?

• 95% chance that ( is contained in the confidence interval

• The probability that the interval contains ( is 95%

• Constructing intervals with this method will produce intervals that contain ( 95% of the time

Steps for a Confidence Interval:

1)

2)

3)

1. a) A Gallup Poll found that 38% of a random sample of 1012 adults said that they believe in ghosts. Find a 95% confidence interval for the true proportion of all adults who believe in ghosts.

b) Construct a 90% confidence interval for the same data.

c) Construct a 99% confidence interval for the same data.

d) How does the confidence level impact the width of the interval?

( How can we make the margin of error smaller?

2. a) A survey of a random sample of 1000 college freshmen found that 47% carry a credit card balance from month to month. Construct a 90% confidence interval for the proportion of all college freshmen who carry a credit card balance from month to month.

b) Does this interval provide evidence that the proportion of college freshmen who carry a balance is below 50%?

3. In a nationwide survey, among a random sample of 526 businesses, 137 reported firing workers for misuse of the internet. Construct a 99% confidence interval for the true proportion of U.S. businesses who have fired workers for misuse of the internet.

Necessary Sample Size:

What p-hat (p) do we use in a sample size calculation?

4. Gallup plans to survey Americans to determine the proportion who approve of attempts to clone humans. What sample size is necessary to be within 4% of the true proportion with 95% confidence?

Sampling Distributions for Means

Population: length of fish (in inches) in my pond.

( Distribution: 2, 7, 10, 11, 14

[pic]________ [pic]_________

Let’s take samples of size 2 (n = 2) from this population:

Pairs | | | | | | | | | | | |[pic] | | | | | | | | | | | |

[pic]_________ [pic]_________

Repeat with sample size n = 3:

Triples | | | | | | | | | | | |[pic] | | | | | | | | | | | |

[pic]_________ [pic]_________

( What do you notice?

General Properties:

Rule 1:

Rule 2:

Rule 3:

Rule 4:

( CLT can be safely applied when ___________

1) 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?

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?

2) A team of biologists has been studying a fishing pond. Let X = the length of a single trout taken at random from the pond. The biologists have determined that X has a normal distribution with mean 10.2 in. and standard deviation 1.4 in.

a) What is the probability that a single trout taken at random from the pond is between 8 and 12 inches long?

b) What is the probability that the mean length of five trout taken at random is between 8 and 12 inches long?

c) What sample mean would be at the 95th percentile (when n = 5)?

3) A soft-drink bottler claims that, on average, cans contain 12 oz of pop. Let X = actual volume of pop in a randomly selected can. Suppose X is normally distributed with ( = .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 exceeds 12.1 oz?

4) Koegels asserts that its “Viennas” brand has an average fat content of 18 grams per hot dog with standard deviation 1 gram. Consumers of this brand would probably not be disturbed if the mean were 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 of getting a sample mean of 18.4 grams or higher?

Does this result indicate that the Koegels claim is incorrect?

Confidence Interval for a Mean:

Conditions for Means:

1. A test for the level of potassium in blood is not perfectly precise. Repeated measurements for the same person on different days vary normally. A random sample of three patients has a mean of 3.2 and a standard deviation of 0.2. What is a 90% confidence interval for the true mean potassium level?

Student’s t-distribution:

How does t compare to normal?

How to find t-values:

Find these t*:

90% confidence when n = 5

95% confidence when n = 15

t-Confidence Interval Formula:

Back to #1 from before:

2. A random sample of 7 high school students yields the following SAT scores:

950 1130 1260 1090 1310 1420 1190

Construct and interpret a 95% confidence interval for the true mean SAT score.

3. A random sample of 50 high school students has a mean SAT score of 1250, with a standard deviation of 105. Find a 95% confidence interval for the mean SAT score of all high school students.

4. The heights of GBHS male students are normally distributed with ( = 2.5 inches. How large a sample is necessary to estimate the true mean height within 0.75 inches with 95% confidence?

Robustness:

5. A medical researcher measured the pulse rate of a random sample of 20 adults and found a mean pulse of 72.69 beats per minute with a standard deviation of 3.86 beats per minute. Assume pulse rate is normally distributed. Compute a 95% confidence interval for the true mean pulse rates of adults.

Another medical researcher claims that the true mean pulse rate for adults is 72 beats per minute. Does our evidence refute this claim? Explain.

6. Consumer Reports tested 14 randomly selected brands of vanilla yogurt and found the following numbers of calories per serving:

160 200 220 230 120 180 140

130 170 190 80 120 100 170

Compute a 98% confidence interval for the average calorie content per serving of vanilla yogurt.

A diet guide claims you get 120 calories from a serving of vanilla yogurt. What does our evidence indicate?

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_____% of the time, the intervals we construct this way will contain the true parameter.

We are _____% confident that the true proportion of context is between _______ and ______.

If we made lots of intervals this way, _____% of them would contain the true proportion.

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