Class Six Turn In: Chapter 15: 30, 32, 38, 44, 48, 50 Chapter 17: 28, 38, 44 For Class Seven: Chapter 18: 32, 34, 36 Chapter 19: 26, 34, 44 Quiz 3 Read Chapters 20 & 22

Objectives Class Six Compute confidence intervals and perform significance tests regarding the means of two distinct populations. Recognize which confidence interval and significance tests are appropriate for a given problem. Discuss the requirements for using t-distributions in confidence intervals and significance tests. Compute confidence intervals and perform significance tests for proportions for one sample problems.

Two Sample Problems The goal of inference is to compare the responses to two treatments or to compare the characteristics of two populations. We have separate sample from each treatment or each population. We can compare either the centers or the spreads of the two groups in a two sample setting.

Conditions for Comparing Two Population Means We have two SRSs from two distinct populations. –the samples are independent i.e. one sample has no influence over the other –we measure the same variable for each sample both populations are Normally distributed –the means and the standard deviations of the populations are unknown –the distributions have similar shapes and there are no strong outliers

Two Sample t Procedures The standard deviation of the observed differences is given by the formula: –this value gets larger as there is more variation in either sample i.e. σ 1 or σ 2 gets larger –this value gets smaller as the sample sizes n 1 and n 2 get larger Since we do not know the standard deviations of the populations we estimate them by using the standard deviations of the samples. The result is our standard error given by the formula: standardizing the difference gives us the two sample t statistic

Two Sample t Distributions the two sample t statistic has approximately a t distribution it does not have exactly a t distribution even if the two sample populations are exactly Normal the degrees of freedom is calculated by a complex formula –the degrees of freedom need not be a whole number –there are two options for using the two sample t procedures option 1: with software, use the t statistic with accurate critical values from the approximating t distribution option 2: without software, use the statistic t with critical values from the t distribution with degrees of freedom equal to the smaller of n 1 – 1 or n 2 – 1.

Two Sample Confidence Intervals Draw an SRS of size n 1 from a Normal population with unknown mean μ 1, and draw an independent SRS of size n 2 from another Normal population with unknown mean μ 2. A confidence interval for μ 1 - μ 2 is given by: where t* is the critical value for the t(k) density curve with area C between – t* and t*. –the degrees of freedom k are equal to the smaller of n 1 – 1 or n 2 – 1 –this interval has confidence level at least C no matter what the population standard deviation may be.

Two Sample Hypothesis Testing Draw an SRS of size n 1 from a Normal population with unknown mean μ 1, and draw an independent SRS of size n 2 from another Normal population with unknown mean μ 2. To test the hypothesis H 0 : μ 1 = μ 2 calculate the two sample t statistic using the formula: –use p values or critical values for the t(k) distribution –the true p value or fixed significance level will always be equal to or less than the value calculated from t(k) no matter what values the unknown population standard deviations have.

Robustness of the Two Sample t Procedures two sample t procedures are more robust than one sample t methods especially when the distributions are not symmetric when sizes of samples are equal and the two populations have distributions with similar shapes the probabilities from the t table are quite accurate even for samples as small as n = 5 adapt the rules for one sample t procedures by replacing sample size n with sum of the sample sizes n 1 + n 2.

Cautions About Two Sample t Procedures always choose the two sample t statistic procedures for populations with unequal variances –this test is valid whether or not the population variances are equal –the other choice assumes the variances are equal and averages (pools) the two sample variance to obtain an estimate of the common population variance this results in a pooled two sample t statistic it is equal to out t statistic only if the two sample sizes are the same it has the t distribution with exactly n 1 + n 2 – 2 degrees of freedom if the two population variances are equal and the population distributions are exactly Normal do not use t procedures to make inferences about the population standard deviations

F Tests for Comparing Two Standard Deviations given that you have two independent SRSs from two Normal populations, a sample of size n 1 from N(μ 1, σ 1 ) and a sample of size n 2 from N(μ 2, σ 2 ). the population means and standard deviations are all unknown to test the hypothesis of equal spread, H 0 : σ 1 = σ 2 vs H a : σ 1  σ 2 we use the ratio of sample variances known as the F statistic. –when s 1 2 and s 2 2 are sample variances from independent SRSs of sizes n 1 and n 2 drawn from Normal populations, the F statistic has the F distribution with n 1 – 1 and n 2 – 1 degrees of freedom when H 0 : σ 1 = σ 2 is true.

A study if performed to compare the mean resting pulse rate of adult subjects who regularly exercise to the mean resting pulse rate of those who do not regularly exercise. nmeanstd. dev. Exercisers Nonexercisers

The Sample Proportion p: the parameter that represents the proportion of the population that has some outcome : the statistic that estimates the parameter p.

Sampling Distribution of a Sample Proportion As the sample size increases, the sampling distribution of becomes approximately Normal. The mean of the sampling distribution is p. The standard deviation of the sampling distribution is

Conditions for Inference About a Proportion The data are an SRS from the population of interest, the most important condition. The population is at least 10 times as large as the sample. This ensures that the standard deviation of p-hat is close to the standard deviation of the sampling proportion. The sample size n is large enough to ensure that the distribution of z is close to standard Normal. –large sample confidence interval: use this only when the counts of successes and failures in the sample are both at least 15 –plus four confidence interval: use this when C is at least 90% and the sample size n is at least 10. –significance tests for a proportion: use this test when the sample size n is so large that both np 0 and n(1 – p 0 ) are 10 or more.

The z Statistic standardizing results in the z statistic: to test the hypothesis H 0 : p = p 0 we use the test statistic: to obtain a confidence interval for p we use:

The z Statistic because we don’t know the value of p we replace the standard deviation with the standard error of which gives us a large sample confidence interval of:

The z Statistic to obtain a more accurate confidence interval we use the plus four method replacing the parameter p with the plus four estimate of p known as: giving us the plus four confidence interval:

Choosing a Sample Size The level C confidence interval for a population proportion p will have margin of error approximately equal to a specified value m when the sample size is: where p* is a guessed value for the sample proportion. The margin of error will be less than or equal to m if you take the guess p* to be 0.5 You may also use a guess for p* based on a pilot study or on past experience with similar studies.

A certain soft drink bottler wants to estimate the proportion of its customers that drink another brand of soft drink on a regular basis. A random sample of 100 customers yielded 18 who did in fact drink another brand of soft drink on a regular basis. Compute a 95% confidence interval to estimate the proportion of interest.

Suppose a certain soft drink bottler wants to estimate the proportion of its customers that drink another brand of soft drink on a regular basis using a 99% confidence interval, and we are instructed to do so such that the margin of error does not exceed 1 percent (0.01). What sample size will be required to enable us to create such an interval?

In a 1994 survey of 474 respondents the proportion of respondents who did not spank or hit was 51%. Is this evidence that a majority of the population did not spank or hit?

Objectives Class Six Compute confidence intervals and perform significance tests regarding the means of two distinct populations. Recognize which confidence interval and significance tests are appropriate for a given problem. Discuss the requirements for using t-distributions in confidence intervals and significance tests. Compute confidence intervals and perform significance tests for proportions for one sample problems.

Next Week Class Seven To Be Completed Before Class Seven: Chapter 18: 32, 34, 36 Chapter 19: 26, 34, 44 Quiz 3 Read Chapters 20 & 22