Exam 3: Review of Statistics

The standard error for the difference in means if two random variables M and W , when the two population variances are different, is a. $\sqrt { \frac { s _ { M } ^ { 2 } + s _ { W } ^ { 2 } } { n _ { M } + n _ { W } } }$ . b. $\frac { s _ { M } } { n _ { M } } + \frac { s _ { W } } { n _ { W } }$ . c. $\sqrt { \frac { 1 } { 2 } \left( \frac { s _ { M } ^ { 2 } } { n _ { M } } + \frac { s _ { W } ^ { 2 } } { n _ { W } } \right) }$ . d. $\sqrt { \frac { s _ { M } ^ { 2 } } { n _ { M } } + \frac { s _ { W } ^ { 2 } } { n _ { W } } }$ .

Math SAT scores $( Y )$ are normally distributed with a mean of 500 and a standard deviation of 100. An evening school advertises that it can improve students' scores by roughly a third of a standard deviation, or 30 points, if they attend a course which runs over several weeks. (A similar claim is made for attending a verbal SAT course.) The statistician for a consumer protection agency suspects that the courses are not effective. She views the situation as follows: $H _ { 0 } : \mu _ { Y } = 500$ vs. $H _ { 1 } : \mu _ { Y } = 530$ . (a)Sketch the two distributions under the null hypothesis and the alternative hypothesis.

An estimator $\hat { \mu } _ { r }$ of the population value $\mu _ { Y }$ is more efficient when compared to another estimator $\tilde { \mu } _ { Y }$ , if

Degrees of freedom a. in the context of the sample variance formula means that estimating the mean uses up some of the information in the data. b. is something that certain undergraduate majors at your university/college other than economics seem to have an $\infty$ amount of. c. are $( n - 2 )$ when replacing the population mean by the sample mean. d. ensure that $s _ { Y } ^ { 2 } = \sigma _ { Y } ^ { 2 }$ .

Your textbook states that when you test for differences in means and you assume that the two population variances are equal, then an estimator of the population variance is the following "pooled" estimator: $s _ { \text {pooled } } ^ { 2 } = \frac { 1 } { n _ { m } + n _ { w } - 2 } \left[ \sum _ { i = 1 } ^ { n _ { m } } \left( Y _ { i } - \bar { Y } _ { m } \right) ^ { 2 } + \sum _ { i = 1 } ^ { n _ { w } } \left( Y _ { i } - \bar { Y } _ { w } \right) ^ { 2 } \right]$ 2 Explain why this pooled estimator can be looked at as the weighted average of the two variances.

IQs of individuals are normally distributed with a mean of 100 and a standard deviation of 16.If you sampled students at your college and assumed, as the null hypothesis, that they had the same IQ as the population, then in a random sample of size (a) $n = 25$ , find $\operatorname { Pr } ( \bar { Y } < 105 )$ . (b) $n = 100$ , find $\operatorname { Pr } ( \bar { Y } > 97 )$ . (c) $n = 144$ , find $\operatorname { Pr } ( 101 < \bar { Y } < 103 )$ .

The standard error of $\bar { Y } , S E ( \bar { Y } ) = \hat { \sigma } _ { \bar { Y } }$ is given by the following formula:

A scatterplot a. shows how $Y$ and $X$ are related when their relationship is scattered all over the place. b. relates the covariance of $X$ and $Y$ to the correlation coefficient. c. is a plot of $n$ observations on $X _ { i }$ and $Y _ { i }$ , where each observation is represented by the point $\left( X _ { i } , Y _ { i } \right)$ . d. shows $n$ observations of $Y$ over time.

A type I error is

Adult males are taller, on average, than adult females.Visiting two recent American Youth Soccer Organization (AYSO)under 12 year old (U12)soccer matches on a Saturday, you do not observe an obvious difference in the height of boys and girls of that age.You suggest to your little sister that she collect data on height and gender of children in 4th to 6th grade as part of her science project.The accompanying table shows her findings. (a)Let your null hypothesis be that there is no difference in the height of females and males at this age level.Specify the alternative hypothesis.

Consider two estimators: one which is biased and has a smaller variance, the other which is unbiased and has a larger variance.Sketch the sampling distributions and the location of the population parameter for this situation.Discuss conditions under which you may prefer to use the first estimator over the second one.

Your textbook mentions that dividing the sample variance by n -1 instead of n is called a degrees of freedom correction.The meaning of the term stems from the fact that one degree of freedom is used up when the mean is estimated.Hence degrees of freedom can be viewed as the number of independent observations remaining after estimating the sample mean. Consider an example where initially you have 20 independent observations on the height of students.After calculating the average height, your instructor claims that you can figure out the height of the 20th student if she provides you with the height of the other 19 students and the sample mean.Hence you have lost one degree of freedom, or 29 there are only 19 independent bits of information.Explain how you can find the height of the 20th student.

The correlation coefficient

U.S.News and World Report ranks colleges and universities annually.You randomly sample 100 of the national universities and liberal arts colleges from the year 2000 issue.The average cost, which includes tuition, fees, and room and board, is $23,571.49 with a standard deviation of $7,015.52. (a)Based on this sample, construct a 95% confidence interval of the average cost of attending a university/college in the United States.

To derive the least squares estimator $\mu _ { Y }$ , you find the estimator m which minimizes

An estimator $\hat { \mu } _ { Y }$ of the population value $\mu _ { Y }$ is consistent if

Your textbook defined the covariance between X and Y as follows: $\frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) \left( Y _ { i } - \bar { Y } \right)$ Prove that this is identical to the following alternative specification: $\frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } X _ { i } Y _ { i } - \frac { n } { n - 1 } \bar { X } \bar { Y }$

The power of the test a. is the probability that the test actually incorrectly rejects the null hypothesis when the null is true. b. depends on whether you use $\bar { Y }$ or $\bar { Y } ^ { 2 }$ for the $t$ -statistic. c. is one minus the size of the test. d. is the probability that the test correctly rejects the null when the alternative is true.

The formula for the sample variance is

The p -value is defined as follows: