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Statistics 400 - Lecture 11 Today: Finish 8.4; begin Chapter 9 Assignment #4: 8.54, 8.103, 8.104, 9.20, 9.30 Not to be handed in Next week…Case Studies and Review No Lab Friday and Next Tuesday Example A company wishes to monitor customer service A senior manager claim that the average waiting time is 3 minutes The waiting times of 75 randomly selected calls to a customer service hotline were recorded The calls had a sample mean of 3.4 minutes and sample standard deviation of 2.4 minutes Test this claim with a significance level of 0.05 Types of Errors What is the probability of rejecting H0 when it is indeed true? If H0 is true, how often do we make the right decision? Have only considered probability of rejecting when H0 is true Suppose H0 is not true. Ideally, what happens? Have 2 types of error: Type I error: Reject H0 when H0 is true Type II error: Fail to Reject H0 when H0 is false The probability of a type I error is: Probability of type II error is: Power of a Test is: Small Sample Inference for Normal Populations Do not always have large samples When is it reasonable to use the Z -test statistic with an unknown standard deviation? X S/ n This statistic has a different sampling distribution called the Student’s t-distribution Student’s t-Distribution Have a random sample of size n from a normal population The distribution of the sample mean is: Distribution of Z is: When population standard deviation is unknown, it is estimated by the sample standard deviation Will use T X as our test statistic S/ n Student’s t-Distribution If x1, x2, …, xn is a random sample from a normal population with mean , and standard deviation , then t X S/ n has a t-distribution with (n-1) degrees of freedom t-distributions are: Bell shaped Symmetric They are not exactly like normal distributions. Why? Probabilities for the t-distribution How many different t-distributions are there? Table 4 can be used for computing probabilities for various t-distributions The probability being computed is a “greater than” probability Example Suppose T has a Student’s t-distribution with 10 degrees of freedom P(T>2.228)= P(T<2.764)= P(T>0)= P(T>2.5)= What is the 90th percentile of this distribution? What is the 10th percentile of this distribution? Small Sample Confidence Interval for the Population Mean If x1, x2, …, xn is a random sample from a normal population with mean , and standard deviation , then a 100(1 )% confidence interval for the population mean is: S X t / 2 n If you have use a distribution instead! Example: Heights of males are believed to be normally distributed Random sample of 25 adult males is taken and the sample mean & standard deviation are 69.72 and 4.15 inches respectively Find a 95% confidence interval for the mean Small Sample Hypothesis Test for the Population Mean Have a random sample of size n ; x1, x2, …, xn H 0 : 0 Test Statistic: t X S/ n Small Sample Hypothesis Test for the Population Mean (cont.) P-value depends on the alternative hypothesis: H1 : 0 : p - value P(T t ) H1 : 0 : p - value P(T t ) H1 : 0 : p - value 2P(T | t |) Where T represents the t-distribution with (n-1 ) degrees of freedom Example: An ice-cream company claims its product contains 500 calories per pint on average To test this claim, 24 of the company’s one-pint containers were randomly selected and the calories per pint measured The sample mean and standard deviation were found to be 507 and 21 calories At the 0.01 level of significance, test the company’s claim What assumptions do we make when using a t-test? How can we check assumptions? Can use t procedures even when population distribution is not normal. Why? Practical Guidelines for t-Tests n<15: Use t procedures if the data are normal or close to normal n<15: If the data are non-normal or outliers are present DO NOT use t procedures n>15: t procedures can be used except in the presence of outliers or strong skewness t>30: t procedures tend to perform well