The area above 5 is shaded blue. This judgment is based on whether the observed difference is beyond what one would expect by chance. Recall from the relevant section in the chapter on sampling distributions that the formula for the standard error of the difference between means is: In order to estimate this quantity, we Using either a Z table or the normal calculator, the area can be determined to be 0.934.

Figure 2. Thus, our estimate of variance is computed using the following formula: where MSE is our estimate of σ2. For now, suffice it to say that small-to-moderate violations of assumptions 1 and 2 do not make much difference. Find the margin of error.

When the sample sizes are small (less than 40), use a t score for the critical value. In contrast, when comparing two independent samples in this fashion the confidence interval provides a range of values for the difference. Since we are assuming the two population variances are the same, we estimate this variance by averaging our two sample variances. return to top | previous page | next page Content ©2016.

The reformatted version of the data in Table 2 is shown in Table 3. Find standard error. The Probability and Statistics Tutor - 10 Hour Course - 3 DVD Set - Learn By Examples!List Price: $39.99Buy Used: $24.90Buy New: $39.99Cracking the AP Statistics Exam, 2014 Edition (College Test In other words, what is the probability that the mean height of girls minus the mean height of boys is greater than 0?

The point estimate for the difference in population means is the difference in sample means: The confidence interval will be computed using either the Z or t distribution for the selected The distribution of the differences between means is the sampling distribution of the difference between means. But first, a note on terminology. Online Calculator: t distribution Formatting Data for Computer Analysis Most computer programs that compute t tests require your data to be in a specific form.

The key steps are shown below. Remember the Pythagorean Theorem in geometry? The approach that we used to solve this problem is valid when the following conditions are met. Men Women Characteristic N s n s Systolic Blood Pressure 1,623 128.2 17.5 1,911 126.5 20.1 Diastolic Blood Pressure 1,622 75.6 9.8 1,910 72.6 9.7 Total Serum Cholesterol 1,544 192.4 35.2

The formula looks easier without the notation and the subscripts. 2.98 is a sample mean, and has standard error (since SE= ). Figure 1. Therefore, the degrees of freedom is 16 + 16 = 32. The last step is to determine the area that is shaded blue.

These formulas, which should only be used under special circumstances, are described below. However,we will first check whether the assumption of equality of population variances is reasonable. All Rights Reserved. The standard error of the difference is 0.641, and the margin of error is 1.26 units.

Note also that this 95% confidence interval for the difference in mean blood pressures is much wider here than the one based on the full sample derived in the previous example, As before, the problem can be solved in terms of the sampling distribution of the difference between means (girls - boys). For men, the average expenditure was $20, with a standard deviation of $3. This is equal to (n1 - 1) + (n2 - 1), where n1 is the sample size of the first group and n2 is the sample size of the second group.

Computationally, this is done by computing the sum of squares error (SSE) as follows: where M1 is the mean for group 1 and M2 is the mean for group 2. If a subject provides two scores, then the scores are not independent. We present a summary of the situations under which each method is recommended. When the sample size is large, you can use a t statistic or a z score for the critical value.

However, the gender difference in this particular sample is not very important. Select a confidence level. Thus, x1 - x2 = $20 - $15 = $5. We could begin by computing the sample sizes (n1 and n2), means ( and ), and standard deviations (s1 and s2) in each sample.

Your cache administrator is webmaster. Based on the confidence interval, we would expect the observed difference in sample means to be between -5.66 and 105.66 90% of the time. When we assume that the population variances are equal or when both sample sizes are larger than 50 we use the following formula (which is also Formula 9.7 on page 274 The standard error of the point estimate will incorporate the variability in the outcome of interest in each of the comparison groups.

Nonetheless it is not inconceivable that the girls' mean could be higher than the boys' mean. Because the sample sizes are large enough, we express the critical value as a z score. Problem 2: Large Samples The local baseball team conducts a study to find the amount spent on refreshments at the ball park. The difference between the two sample means is 2.98-2.90 = .08.

Elsewhere on this site, we show how to compute the margin of error when the sampling distribution is approximately normal. Test Your Understanding Problem 1: Small Samples Suppose that simple random samples of college freshman are selected from two universities - 15 students from school A and 20 students from school We are 95% confident that the difference in mean systolic blood pressures between men and women is between -25.07 and 6.47 units. Generally, the sampling distribution will be approximately normally distributed when the sample size is greater than or equal to 30.

The SE of the difference then equals the length of the hypotenuse (SE of difference = ). Please answer the questions: feedback Stat Trek Teach yourself statistics Skip to main content Home Tutorials AP Statistics Stat Tables Stat Tools Calculators Books Help Overview AP statistics Statistics and A later section describes how to test for differences between the means of two conditions in designs where only one group of subjects is used and each subject is tested in If you cannot assume equal population variances and if one or both samples are smaller than 50, you use Formula 9.9 (in the "Closer Look 9.1" box on page 286) in

From the t-Table t=2.306. Lane Prerequisites Sampling Distribution of Difference between Means, Confidence Intervals, Confidence Interval on the Difference between Means, Logic of Hypothesis Testing, Testing a Single Mean Learning Objectives State the assumptions for There is a second procedure that is preferable when either n1 or n2 or both are small. Our best estimate of the difference, the point estimate, is 1.7 units.

The correct z critical value for a 95% confidence interval is z=1.96. Group 1 Group 2 3 2 4 6 5 8 Here there are two groups, each with three observations. The next section presents sample problems that illustrate how to use z scores and t statistics as critical values. EdwardsList Price: $24.99Buy Used: $3.14Buy New: $17.12Cracking the AP Statistics Exam, 2015 Edition (College Test Preparation)Princeton ReviewList Price: $19.99Buy Used: $0.01Buy New: $11.99Intermediate Statistics For DummiesDeborah J.

Can this estimate miss by much? Each population is at least 20 times larger than its respective sample. Men have lower mean total cholesterol levels than women; anywhere from 12.24 to 17.16 units lower.