Math 37 - Lecture 16

Math 37 - Lecture 16

Confidence Intervals (6.1, 8.1)

Goal: Make a conclusion about the population and use probability to express the strength of our conclusion.

Confidence Interval - Estimate the value of the population parameter.

Procedure:

Have a statistic that is an unbiased estimator of the unknown population parameter, e.g. for p
Use information about the sampling distribution of the statistic to get an idea of how close the value of the statistic is to the population parameter.

If we took many samples from the population, how would the statistic vary?

E.g. If the statistic follows a Normal distribution, can say that 95% of the time the statistic will be within two standard deviations of the parameter. Thus, in 95% of samples, the parameter will be within 2 standard deviations of the statistic.

Proportions - The sampling distribution of is approximately Normal (Normal approximation to the Binomial) with mean p and standard deviation . In 95% of samples, p will be in the interval ( -2, +2).

Estimate SD() by plugging in for p.

Example In June 1994, the cable television show Nick-at-Nite conducted a phone-in survey of its viewers asking: "Who had more magic powers - Jeannie or Samantha"? Samantha received 810,000 votes to Jeannie's 614,000.

- What's the population parameter?

- Find a 95% confidence interval for the population parameter

- What do you think about the width of this interval? Does this make sense?

Confidence interval indicates plausible values for the parameter based on the observed data.

Means - The sampling distribution of is Normal (if population is) or approximately Normal (Central Limit Theorem) with mean m an standard deviation . In 95% of samples, will be within 2 of m. In 95% of the samples m will be within 2 of , that is, in the interval: .

Example In a 1993 study, researchers took a sample of 25 people who claimed to have an intense experience with an unidentified flying object (UFO) and a sample of 25 people who did not claim to have such an experience. They then compared the two groups on a variety of variables, including IQ. The sample mean IQ of the UFO group was 101.6 and the standard deviation of these IQ's was 8.9

- Construct a 95% confidence interval for the population mean IQ of all people who have had intense UFO experiences.

- Explain why the confidence interval is not so narrow.

What if we wanted to be 90% confident? 97% confident?

In general - For any Normal distribution, given any value C, can find a number, z*, such that the Normal distribution has probability C lying within z* standard deviations of its mean: use Table A to find z* such that the interval (-z*,z*) has probability (1-C)/2 on either side.

Critical Value, z*

Let z* be the point in Standard Normal distribution with probability p lying to its right. Call z* the (upper) critical value.