Statistics Lecture 11

zToday: Finish 8.4; begin Chapter 9 zAssignment #4: 8.54, 8.103, 8.104, 9.20, 9.30 zNot to be handed in zNext week…Case Studies and Review zNo Lab Friday and Next Tuesday

Example zA company wishes to monitor customer service zA senior manager claim that the average waiting time is 3 minutes zThe waiting times of 75 randomly selected calls to a customer service hot- line were recorded zThe calls had a sample mean of 3.4 minutes and sample standard deviation of 2.4 minutes zTest this claim with a significance level of 0.05

Types of Errors zWhat is the probability of rejecting H 0 when it is indeed true? zIf H 0 is true, how often do we make the right decision? zHave only considered probability of rejecting when H 0 is true zSuppose H 0 is not true. Ideally, what happens?

zHave 2 types of error: yType I error: Reject H 0 when H 0 is true yType II error: Fail to Reject H 0 when H 0 is false zThe probability of a type I error is: zProbability of type II error is: zPower of a Test is:

Small Sample Inference for Normal Populations zDo not always have large samples zWhen is it reasonable to use the Z -test statistic with an unknown standard deviation? zThis statistic has a different sampling distribution called the Student’s t-distribution

Student’s t-Distribution zHave a random sample of size n from a normal population zThe distribution of the sample mean is: zDistribution of Z is: zWhen population standard deviation is unknown, it is estimated by the sample standard deviation zWill use as our test statistic

Student’s t-Distribution zIf x 1, x 2, …, x n is a random sample from a normal population with mean, and standard deviation, then has a t-distribution with (n-1) degrees of freedom

zt-distributions are: yBell shaped ySymmetric zThey are not exactly like normal distributions. Why?

Probabilities for the t-distribution zHow many different t-distributions are there? zTable 4 can be used for computing probabilities for various t-distributions zThe probability being computed is a “greater than” probability

Example zSuppose T has a Student’s t-distribution with 10 degrees of freedom yP(T>2.228)= yP(T<2.764)= yP(T>0)= yP(T>2.5)= zWhat is the 90 th percentile of this distribution? zWhat is the 10 th percentile of this distribution?

Small Sample Confidence Interval for the Population Mean zIf x 1, x 2, …, x n is a random sample from a normal population with mean, and standard deviation, then a confidence interval for the population mean is: zIf you have use a distribution instead!

Example: zHeights of males are believed to be normally distributed zRandom sample of 25 adult males is taken and the sample mean & standard deviation are and 4.15 inches respectively zFind a 95% confidence interval for the mean

Small Sample Hypothesis Test for the Population Mean zHave a random sample of size n ; x 1, x 2, …, x n z zTest Statistic:

Small Sample Hypothesis Test for the Population Mean (cont.) zP-value depends on the alternative hypothesis: y zWhere T represents the t-distribution with (n-1 ) degrees of freedom

Example: zAn ice-cream company claims its product contains 500 calories per pint on average zTo test this claim, 24 of the company’s one-pint containers were randomly selected and the calories per pint measured zThe sample mean and standard deviation were found to be 507 and 21 calories zAt the 0.01 level of significance, test the company’s claim

zWhat assumptions do we make when using a t-test? zHow can we check assumptions? zCan use t procedures even when population distribution is not normal. Why?

Practical Guidelines for t-Tests zn<15: Use t procedures if the data are normal or close to normal zn<15: If the data are non-normal or outliers are present DO NOT use t procedures zn>15: t procedures can be used except in the presence of outliers or strong skewness zt>30: t procedures tend to perform well