The Practice of StatisticsThird Edition

Chapter 10:Estimating with Confidence

Copyright © 2008 by W. H. Freeman & Company

Daniel S. Yates

Ex.

Suppose a sample of 50 men had a mean score of 109 on an intelligence test.

• We can estimate that the population mean , is approximately 109.

• x bar is normally distributed.

• The mean of the sampling distribution is equal to the unknown population mean.

• The standard deviation of x bar for an SRS of 50 given the population standard deviation is 15/(50)0.5 = 2.1

• The 68 – 95 – 99.7 rule states that about 95% of all possible sample means x bar will be within 2 standard deviations of the population mean

• In 95% of all possible samples the unknown , lies between x bar + or – 4.2

• We are 95% confident that lies between 109 + 4.2; that is (104.8 , 113.2)

• There are only two possibilities:

1. The interval between 104.8 and 113.2 contains the true population mean

2. Our SRS was one of the few samples for which x bar is not within 4.2 points of the true Only 5% of all samples give such inaccurate results.

The method we used gives the correct result 95% of the time.

Applet showing confidence intervals:

http://onlinestatbook.com/stat_sim/conf_interval/index.html

Suppose you want to construct an 80% confidence interval

Confidence

levelTail area Z*

80% 0.1 1.282

90% 0.05 1.645

95% 0.025 1.960

99% 0.005 2.576

Confidence level is usually chosen as > 0.90

Margin of error

estimate

Ex.

A questionnaire of 160 hotel managers asked how long they had been with their current company. The average time was reported as 11.78 years. Give a 99% confidence interval for the mean number of years that the entire population of managers have been with there current company. Assume the standard deviation of the population is years.

11.78 + 2.576(3.2/√160) = 11.78 + 0.652 = (11.128, 12.432)

We are 99% confident that the true population mean lies between 11.128 and 12.432.

The method we used will give the correct result 99% of the time.

Margin of error decreases when;

1) z* gets smaller; but this makes confidence level smaller

) is small – sample drawn from less spread population.

3) n, sample size is large. Quadrupling the sample size cuts margin of error in half.

• Ideally, we would like; 1) high confidence; method almost always gives the right result.

and

2) small margin of error; population parameter estimated very precisely.

How to choose a sample size for a desired margin of error.

Ex.

How many observations must be made to produce results accurate to within + 0.005 with 95% confidence? Assume

z* /√n)n => 7.1 < n ; choose n greater than or equal to 8

You must round up to next integer

It is incorrect to say that the probability is 95% that the true mean lies within a certain interval.

We can say that we are 95% confident that the mean lies within a certain interval or ; The method we used to calculate the interval gives the correct result in 95% of all possible sample of a particular size.

Tests of Significance

• Significance tests assess the evidence provided by the data in favor of some claim about the population.

• Significance tests begin by stating a hypothesis about a population parameter.

• The null hypothesis Ho, is always stated as an equivalence.

Ho : o

• The alternative hypothesis Ha, can be stated in one of three ways.

Ha : ≠ o

< o

> o

Ex.

A car manufacturer claims that one of their car models gets 33mpg. A random sample of 30 cars is selected and the mean gas mileage of this sample x-bar is calculated to be 31 mpg. Can we refute the claim of the automaker? Assume 3.5 mpg.

Ho: 33 mpg

Ha: mpg

x - bar = 31 mpg, sample std. = 3.5/√30 = 0.639

33

3.5/√30 = 0.639

3331 32.361

- 0.639

3331

33

0.00087

• X-bar = 31 is way out on the normal curve. So far out that a result this small almost never occurs by chance if the true 33 mpg.

• This is good evidence that the automakers claim should be rejected in favor of the alternate hypothesis, 33 mpg

• Generally P-values < 0.05 are considered small enough to reject the Ho. It is statistically significant.

P( z < -3.12) = 0.00087

Significance level

• We compare the P – value with a fixed value that we regard as decisive.

• The decisive value of P is called the significance level. Symbol =>

• Choosing = 0.05 require that the data give evidence against Ho so extreme that it would happen in no more than 5% of the possible samples if Ho is true. = 0.01 require that the data give evidence against Ho so extreme that it would happen in no more than 1% of the possible samples if Ho is true.

• If the P – value is as small or smaller than , we say that the data are statistically significant at level The null hypothesis should be rejected in favor of the alternate hypothesis.

If P-value is low,

reject the HO

One sided test

Two sided test

{

Choosing an level in significance tests

• If Ho represents an assumption that people you must convince have believed for a long period of time, strong evidence (small ) is needed to persuade them.

• If the consequences of rejecting Ho are drastic; ie expensive, finality. You may want strong evidence, (small ).

• May be more useful to report the P-value so each individual may decide for themselves.

• Even though significance levels of 0.10, 0.05 and 0.01 have been used traditionally. The border between what levels are significant is not black and white. Not much difference between P-values of 0.049 and 0.051.

• No significance level is sacred.

Inference as decision

Type I and Type II errors

• If we reject Ho (accept Ha) when Ho is really true, this is a Type I error.

• If we reject Ha (accept Ho) when Ha is really true, this is Type II error.

Ho True Ha True

Reject Ho Type I Error

Correct Decision

Reject Ha Correct Decision

Type II error

Significance and Type I error

• The significance level of any fixed level significance test is equal to the probability of making a Type I error.

• the value of is the probability that the test will reject the null hypothesis Ho when Ho is really true.

Power of the test

• The probability that a fixed level a significance test will reject Ho when Ha is true is called the power of the test.

• Increasing sample size n, increases the power of the test.

• Increasing the significance level increases the power of the test.