Exam 3: Review of Statistics

Assume that under the null hypothesis, has an expected value of 500 and a standard deviation of 20.Under the alternative hypothesis,the expected value is 550.Sketch the probability density function for the null and the alternative hypothesis in the same figure.Pick a critical value such that the p-value is approximately 5%.Mark the areas,which show the size and the power of the test.What happens to the power of the test if the alternative hypothesis moves closer to the null hypothesis,i.e. ,, = 540,530,520,etc.?

An estimator of the population value is more efficient when compared to another estimator ,if

During the last few days before a presidential election,there is a frenzy of voting intention surveys.On a given day,quite often there are conflicting results from three major polls. (a)Think of each of these polls as reporting the fraction of successes (1s)of a Bernoulli random variable Y,where the probability of success is Pr(Y = 1)= p.Let be the fraction of successes in the sample and assume that this estimator is normally distributed with a mean of p and a variance of .Why are the results for all polls different,even though they are taken on the same day? (b)Given the estimator of the variance of , ,construct a 95% confidence interval for .For which value of is the standard deviation the largest? What value does it take in the case of a maximum ? (c)When the results from the polls are reported,you are told,typically in the small print,that the "margin of error" is plus or minus two percentage points.Using the approximation of 1.96 ≈ 2,and assuming,"conservatively," the maximum standard deviation derived in (b),what sample size is required to add and subtract ("margin of error")two percentage points from the point estimate? (d)What sample size would you need to halve the margin of error?

If the null hypothesis states H0 : E(Y)= µY,0,then a two-sided alternative hypothesis is

The formula for the sample variance is

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: Explain why this pooled estimator can be looked at as the weighted average of the two variances.

Some policy advisors have argued that education should be subsidized in developing countries to reduce fertility rates.To investigate whether or not education and fertility are correlated,you collect data on population growth rates (Y)and education (X)for 86 countries.Given the sums below,compute the sample correlation: = 1.594; = 449.6; = 6.4697; = 0.03982; = 3,022.76

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 there are only 19 independent bits of information.Explain how you can find the height of the 20th student.

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. Height of Young Boys and Girls,Grades 4-6,in inches (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. (b)Find the difference in height and the standard error of the difference. (c)Generate a 95% confidence interval for the difference in height. (d)Calculate the t-statistic for comparing the two means.Is the difference statistically significant at the 1% level? Which critical value did you use? Why would this number be smaller if you had assumed a one-sided alternative hypothesis? What is the intuition behind this?

The size of the test

A manufacturer claims that a certain brand of VCR player has an average life expectancy of 5 years and 6 months with a standard deviation of 1 year and 6 months.Assume that the life expectancy is normally distributed. (a)Selecting one VCR player from this brand at random,calculate the probability of its life expectancy exceeding 7 years. (b)The Critical Consumer magazine decides to test fifty VCRs of this brand.The average life in this sample is 6 years and the sample standard deviation is 2 years.Calculate a 99% confidence interval for the average life. (c)How many more VCRs would the magazine have to test in order to halve the width of the confidence interval?

To derive the least squares estimator µY,you find the estimator m which minimizes

Let Y be a Bernoulli random variable with success probability Pr(Y = 1)= p,and let Y1,... ,Yn be i.i.d.draws from this distribution.Let be the fraction of successes (1s)in this sample.In large samples,the distribution of will be approximately normal,i.e. , is approximately distributed N(p, ).Now let X be the number of successes and n the sample size.In a sample of 10 voters (n=10),if there are six who vote for candidate A,then X = 6.Relate X,the number of success,to ,the success proportion,or fraction of successes.Next,using your knowledge of linear transformations,derive the distribution of X.

Imagine that you had sampled 1,000,000 females and 1,000,000 males to test whether or not females have a higher IQ than males.IQs are normally distributed with a mean of 100 and a standard deviation of 16.You are excited to find that females have an average IQ of 101 in your sample,while males have an IQ of 99.Does this difference seem important? Do you really need to carry out a t-test for differences in means to determine whether or not this difference is statistically significant? What does this result tell you about testing hypotheses when sample sizes are very large?

The following statement about the sample correlation coefficient is true.

The power of the test

When you are testing a hypothesis against a two-sided alternative,then the alternative is written as

(Requires calculus. )Let Y be a Bernoulli random variable with success probability Pr(Y = 1)= p.It can be shown that the variance of the success probability p is .Use calculus to show that this variance is maximized for p = 0.5.

You have collected data on the average weekly amount of studying time (T)and grades (G)from the peers at your college.Changing the measurement from minutes into hours has the following effect on the correlation coefficient:

When testing for differences of means,you can base statistical inference on the