3 t Procedures for CIDraw an SRS of size n from a population having unknown mean μ and unknown standard deviation σ. A level C confidence interval for μ is:where t* is the upper (1 – C)/2 critical value for thet distribution with (n – 1) degrees of freedom. The interval is exact when the population distribution is normal and is approximately correct for large n in other cases.
4 Conditions for t σ is unknown Sample is an SRS from the population of interest (Very Important).
5 More Conditions for t procedures Because the t-procedures are strongly affected by outliers the sample conditions vary:NormalNot normal(even skewed is ok), but n ≥ 40Medium sample, but stemplot looks fairly normal, no outliers or strong skew n ≥ 15Smaller sample, n < 15, but stemplot looks very normal, no outliers or skewness.
7 U-Try CI ProblemA study of the ability of individuals to walk in a straight line reported the following data on cadence(strides per second) for a sample of n = 20 randomly selected healthy men:Construct and interpret a 99% confidence interval for the population mean cadence.
8 t Hypothesis Test Procedures Same conditions as for CITo test H0: μ = μ0Compute test statistic:Compare the p-value from the t distribution with (n – 1) degrees of freedom against:HA: μ > μ0 is P(T ≥ t)HA: μ < μ0 is P(T ≤ t)HA: μ ≠ μ0 is 2P(T ≥ |t|)
9 Hypothesis Test Example Much concern has been expressed in recent years regarding the practice of using nitrates as meat preservatives. In one study involving possible effects of these chemicals, bacteria cultures were grown in a medium containing nitrates. The rate of uptake of radio-labeled amino acid was then determined for each culture, yielding the following observations:7568
10 Example cont.Suppose it is known that the true average uptake for cultures without nitrates is Do the data suggest that the addition of nitrates results in a decrease in the true average uptake?Test using significance level of .10.
11 U-Try Hypothesis Test Problem A survey of teenagers and parents in Canada conducted by the polling organization Ipsos included questions about internet use. It was reported that for a sample of 534 randomly selected teens, the mean number of hours per week spent online was 14.6 and the standard deviation 11.6.
12 U-try cont.What does the large standard deviation tell you about the distribution of time spent on the internet by teensDo the sample data provide convincing evidence, α = .05, that the mean number of hours that teens spend online is greater than 10 hours per week?
14 Two Sample SituationsMatched Pairs Test: When an experiment is done using a matched pairs design, we apply the t-procedures looking at the differences between the pairs.Although there are two samples, they get combined into one sample of differences.We test whether the difference is 0, or something other than 0.
15 Matched Pairs ExampleCan taking chess lessons and playing chess daily improve memory? Sixth grade students who had not previously played chess participated in a program where they took chess lessons and played chess daily for 9 months. Each student took a memory test before starting the chess program and again at the ned of the 9 month period. We look at the pre-test and post-test score of students.
16 Data Memory Test Score Student Pre- Test Post-Test Difference 1 510 850-3402610790-1803640-2104675775-10056007006550-2257-9086259450690-24010720-55115755403512680-5
17 Chess cont.Do the data give convincing evidence that the chess program significantly increased the memory test score? Use a paired t-test with significance level .05 to carry out the test.