Inference for Proportions Chapter 8 Inference for Proportions

Inference: using results from a random sample to draw conclusions about a population

In this chapter, you will use your understanding of sampling distributions developed in Chapter 7

One basic fact about sampling distributions will be used over and over again

If the sampling distribution can be considered approximately normal, 95% of all the sample means (or sample proportions) will fall within 1.96 standard errors of the population mean (or population proportion).

Estimating a Proportion with Confidence Section 8.1 Estimating a Proportion with Confidence

Reasonably Likely and Rare Events Reasonably likely events are those in the middle 95% of the distribution of all possible outcomes. The outcomes in the upper 2.5% and lower 2.5% of the distribution are rare events - - they happen, but rarely.

Reasonably Likely and Rare Events middle 95%

Reasonably Likely Events Given: (1) Random sampling from a binomial population is used repeatedly

Reasonably Likely Events Given: (1) Random sampling from a binomial population is used repeatedly (2) Both np and n(1- p) are at least 10

Reasonably Likely Events Given: (1) Random sampling from a binomial population is used repeatedly (2) Both np and n(1- p) are at least 10 -- have approx. normal distribution

Reasonably Likely Events Given: (1) Random sampling from a binomial population is used repeatedly (2) Both np and n(1-p) are at least 10 Then, 95% of all sample proportions p will fall within 1.96 standard errors of the population proportion, p.

Reasonably Likely Events or about 95% of sample proportions will fall within the interval where n is the sample size.

Suppose you flip a fair coin 100 times and define heads as success . (1) What are the reasonably likely values of the sample proportion p? (2) What number of heads is reasonably likely?

Suppose you flip a fair coin 100 times. (1) What are the reasonably likely values of the sample proportion p? (2) What number of heads is reasonably likely? What is p? What is n?

Suppose you flip a fair coin 100 times. (1) What are the reasonably likely values of the sample proportion p? (2) What number of heads is reasonably likely? What is p? probability of success is 0.5 What is n?

Suppose you flip a fair coin 100 times. (1) What are the reasonably likely values of the sample proportion p? (2) What number of heads is reasonably likely? What is p? probability of success is 0.5 What is n? 100 flips means sample size is 100

Suppose you flip a fair coin 100 times. (1) What are the reasonably likely values of the sample proportion p? =

Suppose you flip a fair coin 100 times. (1) What are the reasonably likely values of the sample proportion p? = 0.5 0.1 So, reasonable likely values of p are from 0.4 to 0.6

Suppose you flip a fair coin 100 times. (2) What number of heads is reasonably likely?

Suppose you flip a fair coin 100 times. (2) What number of heads is reasonably likely? In about 95% of the samples, the number of successes x in the sample will be in the interval about 40 to 60 heads

Suppose 35% of a population think that they pay too much for car insurance. A polling organization takes a random sample of 500 people from this population and computes the sample proportion, p, of people who think they pay too much for car insurance. (a) There is a 95% chance that p will be between what two values?

Suppose 35% of a population think that they pay too much for car insurance. A polling organization takes a random sample of 500 people from this population and computes the sample proportion, p, of people who think they pay too much for car insurance.

Suppose 35% of a population think that they pay too much for car insurance. A polling organization takes a random sample of 500 people from this population and computes the sample proportion, p, of people who think they pay too much for car insurance. (b) Is the organization reasonably likely to get 145 people in the sample who think they pay too much for car insurance?

(0.308, 0.392) (b) Is the organization reasonably likely to get 145 people in the sample who think they pay too much for car insurance?

(b) Is the organization reasonably likely to get 145 people in the sample who think they pay too much for car insurance?

Suppose 40% of students in your graduating class plan to go on to higher education. You survey a random sample of 50 of your classmates and compute the sample proportion, p, of students who plan to go on to higher education. (a) There is a 95% chance that p will be between what two numbers? (b) Is it reasonably likely to find that 25 students in your sample plan to go on to higher education?

(a) There is a 95% chance that p will be between what two numbers?

(b) Is it reasonably likely to find that 25 students in your sample plan to go on to higher education?

(b) Is it reasonably likely to find that 25 students in your sample plan to go on to higher education? (0.264, 0.536) Getting 25 out of 50 is a sample proportion of 0.5. This is a reasonably likely event from a population with p = 0.4.

Turn to page 470. Read through Activity 8.1a.

Activity 8.1a, Page 470 1. Out of a sample of 40 students, 27 students could make the Vulcan salute with both hands at once. Write-up for this lab is due Monday. Justify your answers—simple yes or no answers earn no credit

A 95% confidence interval consists of those population proportions p for which the sample proportion p is reasonably likely.

A Complete Chart of Reasonably Likely Sample Proportions for n = 40

A Complete Chart of Reasonably Likely Sample Proportions for n = 40

Page 473, D3

Page 473, D3

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Page 473, D4

Page 473, D4

Page 473, D4 No. The horizontal line segment at p = 0.3 goes from about 0.158 to 0.442, so a sample proportion of 0.6 isn’t a reasonably likely result for a population with only 30% men.

Page 473, D5

Page 473, D5

Page 473, D5

Page 473, D5

Page 473, D6

Page 473, D6 The populations for which a sample proportion of 0.5 is reasonably likely are 35% to 65%. This can be written as 50% 15%.

Page 473, D7

Page 473, D7 You don’t need a confidence interval for p because you already know exactly what that is from our sample and you know that it probably would have been different if you had taken a different sample. What you want is an interval that has a good chance of capturing the true but unknown proportion of successes p in the population from which the sample was taken.

A confidence interval for the proportion of successes p in the population is given by the formula Here n is the sample size and p is the proportion of successes in the sample.

Value of z* depends on how confident you want to be that p will be in the confidence interval.

Value of z* depends on how confident you want to be that p will be in the confidence interval. For 90% confidence interval, use 1.645

Value of z* depends on how confident you want to be that p will be in the confidence interval. For 90% confidence interval, use 1.645 For 95% confidence interval, use 1.96

Value of z* depends on how confident you want to be that p will be in the confidence interval. For 90% confidence interval, use 1.645 For 95% confidence interval, use 1.96 For 99% confidence interval, use 2.576

Check Conditions This confidence interval is reasonably accurate when three conditions are met:

Check Conditions This confidence interval is reasonably accurate when three conditions are met: (1) Sample was a simple random sample from a binomial population

Check Conditions This confidence interval is reasonably accurate when three conditions are met: (1) Sample was a simple random sample from a binomial population (2) Both np and n(1 – p) are at least 10

Check Conditions This confidence interval is reasonably accurate when three conditions are met: (1) Sample was a simple random sample from a binomial population (2) Both np and n(1 – p) are at least 10 (3) Size of the population is at least 10 times the size of the sample

The quantity is called the margin of error.

The quantity is called the margin of error.

Questions?