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Comparing Two Means: Understanding Confidence Intervals and Hypothesis Testing

harlan avatar
harlan
18 diapositives · 342 Views · Jul 29, 2014

This section delves into the methodology of comparing two group means through statistical techniques, primarily focusing on boxplots, two-sample t-tests, and construction of confidence intervals. It outlines the assumptions and conditions required for these tests, such as independence and normality, and explains how to interpret results through standardized sampling models. The text also illustrates concepts with examples, guiding readers through the implications of visual cues on consumption behaviors, and emphasizes the importance of random sampling in enhancing the validity of results.

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Comparing Two Means: Understanding Confidence Intervals and Hypothesis Testing
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  1. Solution: A

  2. Section 11.2 Comparing Two Means

  3. The natural display for comparing two groups is boxplots of the data for the two groups, placed side-by-side. For example: Plot the Data

  4. Comparing Two Means • Once we have examined the side-by-side boxplots, we can turn to the comparison of two means. • This time the parameter of interest is the difference between the two means, 1 – 2. • The standard deviation of the difference between two sample means is • We still don’t know the true standard deviations of the two groups, so we need to estimate and use the standard error

  5. Comparing Two Means (cont.) • Because we are working with means and estimating the standard error of their difference using the data, we shouldn’t be surprised that the sampling model is a Student’s t. • The confidence interval we build is called a two-sample t-interval(for the difference in means). • The corresponding hypothesis test is called a two-sample t-test.

  6. Sampling Distribution for the Difference Between Two Means • When the conditions are met, the standardized sample difference between the means of two independent groups can be modeled by a Student’s t-model with a number of degrees of freedom found with a special formula. • We estimate the standard error with

  7. Assumptions and Conditions • Independence Assumption (Each condition needs to be checked for both groups.): • Randomization Condition:Were the data collected with suitable randomization (representative random samples or a randomized experiment)? • 10% Condition:We don’t usually check this condition for differences of means. We will check it for means only if we have a very small population or an extremely large sample.

  8. Assumptions and Conditions (cont.) • Normal Population Assumption: • Nearly Normal Condition: This must be checked for both groups. A violation by either one violates the condition. • Independent Groups Assumption: The two groups we are comparing must be independent of each other. (See Chapter 25 if the groups are not independent of one another…)

  9. TI – Tips Confidence Interval and hypothesis tests for difference in means (from two independent samples) • For confidence intervals … • STAT TESTS 0:2-SampTint • For hypothesis testing … • STAT TESTS 4:2-sampleTTest

  10. Two-Sample t-Interval When the conditions are met, we are ready to find the confidence interval for the difference between means of two independent groups, 1 – 2. The confidence interval is where the standard error of the difference of the means is The critical value t*df depends on the particular confidence level, C, that you specify and on the number of degrees of freedom, which we get from the sample sizes and a special formula.

  11. Degrees of Freedom • The special formula for the degrees of freedom for our t critical value is a bear: • Because of this, we will let technology calculate degrees of freedom for us!

  12. Testing the Difference Between Two Means • The hypothesis test we use is the two-sample t-test for means. • The conditions for the two-sample t-test for the difference between the means of two independent groups are the same as for the two-sample t-interval.

  13. A Test for the Difference Between Two Means • We test the hypothesis H0:1 – 2 = 0, where the hypothesized difference, 0, is almost always 0, using the statistic • The standard error is • When the conditions are met and the null hypothesis is true, this statistic can be closely modeled by a Student’s t-model with a number of degrees of freedom given by a special formula. We use that model to obtain a P-value. • We NEVER pool with a two-sample t-test for means.

  14. Example: Can you tell how much you are eating from how full you are? Or do you need visual cues? Researchers constructed a table with two ordinary 18 oz soup bowls and two identical looking bowls that had been modified to slowly, imperceptibly, refill as they were emptied. They assigned experiment participants to the bowls randomly and served them tomato soup. Those eating from the ordinary bowls had their bowls refilled by ladle whenever they were one-quarter full. If people judged their portions by internal cues, they should eat about the same amount. Ordinary bowl n = 27, y bar = 8.5 oz. s = 6.1 oz. Refilling bowl n = 27, y bar = 14.7 oz. s = 8.4 oz. How much variability do we expect in the difference between the two means?

  15. Example: Researchers randomly assigned people to eat soup from one of two bowls: 27 got ordinary bowls that were refilled by ladle and 27 others were given bowls that secretly refilled as the people ate. The histograms for both groups look unimodal but somewhat skewed to the right. Can the researchers use their data to make inferences about the role of visual cues in determining how much people eat?

  16. Example: What does a 95% confidence interval say about the difference in mean amounts eaten?

  17. Example: Many office “coffee stations” collect voluntary payments for the food consumed. Researchers at the Univ. of Newcastle performed an experiment to see whether the image of eyes watching would change employee behavior. They alternated pictures of eyes looking at the viewer with pictures of flowers each week on the cupboard behind the “honesty box”. They measured the consumption of milk to approximate the amount of food consumed and recorded the contributions each week per liter of milk. Eyes n = 5 weeks, ybar = $.417 per liter, s = $.1811 per liter Flowers n = 5 weeks, ybar = $.1811 per liter, s =$0.67 per liter Do these results provide evidence that there really is a difference in honesty even when it’s only photographs of eyes that are watching?