Student t-Distribution In a randomized comparative experiment on the effects of calcium on blood pressure, researchers divided 54 healthy, white males at random into two groups, takes calcium or placebo. The paper reports a mean seated systolic blood pressure of 114.9 with standard deviation of 9.3 for the placebo group. Assume systolic blood pressure is normally distributed. Can you find a z-interval for this problem? Why or why not? No, don’t know σ Only sample s

Gossett Story

ẋ has a normal distribution Parameters are constant – don’t expect the shape to change, just shift based on changes in ẋ Statistics are variables – each sample s will cause the shape to change away from a normal distribution Can you use s instead of σ when calculating a z-score (so that you can find the +/- 3 σ )? Not exactly. Look at the two equations. ẋ has a normal distribution

Student’s t- distribution Developed by William Gosset Continuous distribution Unimodal, symmetrical, bell-shaped density curve Above the horizontal axis Area under the curve equals 1 Based on degrees of freedom: n-1

T-Curves Basic properties of t-Curves Property 1: The total area under a t-curve equals 1. Property 2: A t-curve extends indefinitely in both directions, approaching the horizontal axis asymptotically Property 3: A t-curve is symmetric about 0.

T-curves continued Property 4: As the number of degrees of freedom becomes larger, t-curves look increasingly like the standard normal curve NOTE: T-Distributions are only used with Inference of MEANS

t- curve vs normal curve Graph examples of t- curve vs normal curve Y1: normalpdf(x) Y2: tpdf(x,2) Y3:tpdf(x,5) use the -0 Change Y3:tpdf(x,30) Window: x = [-4,4] scl =1 Y=[0,.5] scl =1

z – Score and t - Score

Confidence Interval Formula: Standard Error (SD of sampling distribution Critical value estimate Margin of error

How to find Margin of error when σ is not available – find t* Can also use invT on the calculator! For 90% confidence level, 5% is above and 5% is below Need upper t* value with 5% above – so 0.95 is p value invT(p,df) How to find Margin of error when σ is not available – find t* Use Table B for t distributions Look up confidence level at bottom & degrees of freedom (df) on the sides df = n – 1 Find these t* 90% confidence when n = 5 95% confidence when n = 15 t* =2.132 t* =2.145

Assumptions for t-inference Have an SRS from population Check for independent sample σ unknown Normal distribution Given Large sample size Check graph of data

Some Cautions: The data MUST be a SRS from the population (must be random) The formula is not correct for more complex sampling designs, i.e., stratified, etc. No way to correct for bias in data so if bias, data is tossed out

Cautions continued: Outliers can have a large effect on confidence interval Must know σ to do a z-interval – which is unrealistic in practice

Ex. 5 – A medical researcher measured the pulse rate of a random sample of 20 adults and found a mean pulse rate of 72.69 beats per minute with a standard deviation of 3.86 beats per minute. Assume pulse rate is normally distributed. Compute a 95% confidence interval for the true mean pulse rates of adults. One Sample t-interval of means (70.883, 74.497)

Another medical researcher claims that the true mean pulse rate for adults is 72 beats per minute. Does the evidence support or refute this? Explain. The 95% confidence interval contains the claim of 72 beats per minute. Therefore, there is no evidence to doubt the claim.

Example 9: The Wall Street Journal (January 27, 1994) reported that based on sales in a chain of Midwestern grocery stores, President’s Choice Chocolate Chip Cookies were selling at a mean rate of $1323 per week. Suppose a random sample of 30 weeks in 1995 in the same stores showed that the cookies were selling at the average rate of $1208 with standard deviation of $275. Does this indicate that the sales of the cookies is different from the earlier figure?

Assume: Have an SRS of weeks Assume one week’s sale is independent of the next Distribution of sales is approximately normal due to large sample size s unknown H0: m = 1323 where m is the true mean cookie sales per Ha: m ≠ 1323 week Since p-value < a of 0.05, I reject the null hypothesis. There is sufficient evidence to suggest that the sales of cookies are different from the earlier figure.

Example 8: The Degree of Reading Power (DRP) is a test of the reading ability of children. Here are DRP scores for a random sample of 44 third-grade students in a suburban district: (data on note page) At the a = .1, is there sufficient evidence to suggest that this district’s third graders reading ability is different than the national mean of 34?

What are your hypothesis statements? Is there a key word? SRS? Independent Sample? I have an SRS of third-graders Assume population is > 440 Normal? How do you know? Since the sample size is large, the sampling distribution is approximately normally distributed OR Since the histogram is unimodal with no outliers, the sampling distribution is approximately normally distributed Do you know s? What are your hypothesis statements? Is there a key word? s is unknown H0: m = 34 where m is the true mean reading Ha: m = 34 ability of the district’s third-graders One Sample t-Test of Means p-value = 0.5212 a = .1

Compare your p-value to a & make decision Conclusion: Since p-value > a, I fail to reject the null hypothesis. There is not sufficient evidence to suggest that the true mean reading ability of the district’s third-graders is different than the national mean of 34. Write conclusion in context in terms of Ha.