1	Inference for a population mean BPS chapter 18
2	Objectives (BPS chapter 18) Inference about a Population Mean Conditions for inference The t distribution The one-sample t confidence interval Using technology Matched pairs t procedures Robustness of t procedures
3	Conditions for inference about a mean We can regard our data as a simple random sample (SRS) from the population. This condition is very important. Observations from the population have a Normal distribution with mean m and standard deviation s. In practice, it is enough that the distribution be symmetric and single-peaked unless the sample is very small. Both m and standard deviation s are unknown.
4	"Sweetening colas" Sweetening colas Cola manufacturers want to test how much the sweetness of a new cola drink is affected by storage. The sweetness loss due to storage was evaluated by 10 professional tasters (by comparing the sweetness before and after storage): Taster Sweetness loss 1 2.0 2 0.4 3 0.7 4 2.0 5 −0.4 6 2.2 7 −1.3 8 1.2 9 1.1 10 2.3
5	When s is unknown When the sample size is very large, the sample is likely to contain elements representative of the whole population. Then s is a very good estimate of s. But when the sample size is small, the sample contains only a few individuals. Then s is a more mediocre estimate of s.
6	Standard deviation s — standard error of the mean s/√n For a sample of size n, the sample standard deviation s is: n − 1 is the “degrees of freedom.” The value s/√n is called the standard error of the mean SEM. Scientists often present their sample results as the mean ± SEM.
7	The t distributions We test a null and alternative hypotheses with one sample of size n from a normal population N(µ,σ): When s is known, the sampling distribution is normal N(m, s/√n). When s is estimated from the sample standard deviation s, then the sampling distribution follows a t distribution t(m,s/√n) with degrees of freedom n − 1. The value (s/√n) is the standard error of the mean or SEM.
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9	Standardizing the data before using Table C Here, m is the mean (center) of the sampling distribution, and the standard error of the mean s/√n is its standard deviation (width). You obtain s, the standard deviation of the sample, with your calculator.
10	Table C
11	Table A vs. Table C
12	Confidence intervals Reminder: The confidence interval is a range of values with a confidence level C representing the probability that the interval contains the true population parameter. We have a set of data from a population with both m and s unknown. We use to estimate m, and s to estimate s, using a t distribution (df n − 1).
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14	"Red wine" Red wine, in moderation (continued) What is the 95% confidence interval for the average percent change in blood polyphenols?
15	Excel
16	The t-test As in the previous chapter, a test of hypotheses requires a few steps: Stating the null and alternative hypotheses (H₀ versus H_a) Deciding on a one-sided or two-sided test Choosing a significance level a Calculating t and its degrees of freedom Finding the area under the curve with Table C Stating the P-value and interpreting the result
17	Review: test of significance The P-value is the probability, if H₀ is true, of randomly drawing a sample like the one obtained, or more extreme, in the direction of H_a. The P-value is calculated as the corresponding area under the curve, one-tailed or two-tailed depending on H_a:
18	Table C How to:
19	Excel TDIST(x, degrees_freedom, tails) TDIST = p(X > x ), where X is a random variable that follows the t distribution (x positive). Use this function in place of a table of critical values for the t distribution or to obtain the P-value for a calculated, positive t-value. X is the standardized numeric value at which to evaluate the distribution (“t”). Degrees_freedom is an integer indicating the number of degrees of freedom. Tails specifies the number of distribution tails to return. If tails = 1, TDIST returns the one-tailed P-value. If tails = 2, TDIST returns the two-tailed P-value. TINV(probability, degrees_freedom) Returns the t-value of the Student's t-distribution as a function of the probability and the degrees of freedom (for example, t*). Probability is the probability associated with the two-tailed Student’s t distribution. Degrees_freedom is the number of degrees of freedom characterizing the distribution.
20	"Sweetening colas (continued" Sweetening colas (continued) Is there evidence that storage results in sweetness loss for the new cola recipe at the 0.05 level of significance (a = 5%)? H₀: m = 0 versus H_a: m > 0 (one-sided test) the critical value t_a = 1.833 t > t_a thus the result is significant. 2.398< t = 2.70 < 2.821, thus 0.02 > p > 0.01 p < a, thus the result is significant. The t-test has a significant p-value. We reject H₀. There is a significant loss of sweetness, on average, following storage.
21	"Sweetening colas (continued" Sweetening colas (continued)
22	"Red wine" Red wine, in moderation (continued) Does moderate red wine consumption increase the average blood level of polyphenols in healthy men?
23	Matched pairs t procedures Sometimes we want to compare treatments or conditions at the individual level. These situations produce two samples that are not independent — they are related to each other. The members of one sample are identical to, or matched (paired) with, the members of the other sample. Example: Pre-test and post-test studies look at data collected on the same sample elements before and after some experiment is performed. Example: Twin studies often try to sort out the influence of genetic factors by comparing a variable between sets of twins. Example: Using people matched for age, sex, and education in social studies allows us to cancel out the effect of these potential lurking variables.
24	"In these cases" In these cases, we use the paired data to test the difference in the two population means. The variable studied becomes X = x₁ − x₂, and H₀: µ_difference=0; H_a: µ_difference>0 (or <0, or ≠0) Conceptually, this does not differ from tests on one population.
25	"Sweetening colas (revisited" Sweetening colas (revisited) The sweetness loss due to storage was evaluated by 10 professional tasters (comparing the sweetness before and after storage): Taster Sweetness loss 1 2.0 2 0.4 3 0.7 4 2.0 5 −0.4 6 2.2 7 −1.3 8 1.2 9 1.1 10 2.3
26	Does lack of caffeine increase depression? Individuals diagnosed as caffeine-dependent are deprived of all caffeine-rich foods and assigned to receive daily pills. At some time, the pills contain caffeine and at another time they contain a placebo. Depression was assessed. There are two data points for each subject, but we will only look at the difference. The sample distribution appears appropriate for a t-test.
27	Does lack of caffeine increase depression? For each individual in the sample, we have calculated a difference in depression score (placebo minus caffeine). There were 11 “difference” points, thus df = n − 1 = 10. We calculate that = 7.36; s = 6.92
28	Robustness The t procedures are exactly correct when the population is distributed exactly normally. However, most real data are not exactly normal. The t procedures are robust to small deviations from normality. This means that the results will not be affected too much. Factors that do strongly matter are: Random sampling. The sample must be an SRS from the population. Outliers and skewness. They strongly influence the mean and therefore the t procedures. However, their impact diminishes as the sample size gets larger because of the Central Limit Theorem.
29	Reminder: Looking at histograms for normality