Statistical Analysis: Hypothesis Testing and Probability in a Nutshell

1. Hypothesis Testing

1.For each of the following statements, circle the letter “T” if it is true, and “F” if it is false.  

T    F    (a)   When the p-value is < α we reject H0.  

T   F    (b)   A binomial variable is constructed from 50 trials where the probability of success in each trial is 0.99.  The normal approximation to the binomial distribution can be used to estimate the likelihood of obtaining a value of y=49.  

T    F    (c)   When comparing the variances of 2 different populations using sample data,   (s1 / s2)2 follows the F distribution when the parent distributions are normal. 

T     F    (d)   If f0.95,20,10=0.426 then f0.05,10,20=1-0.426=0.574.

T     F    (e)   A confidence interval for the difference in two population means is being constructed using sample data.  The confidence interval will be tighter if it can be assumed that the two populations have equal variances.

T     F   (f)   In regression analysis, the term SSR quantifies the scatter about  that is explained by the regression line.

T     F    (g)   A large value of R2 in linear least squares regression analysis means that the regression equation accurately predicts all yi values.  

T     F    (h)  The number of degrees of freedom in the numerator of the f statistic is n1-2 when testing σ1 vs. σ2

T     F    (i)   A linear least squares regression equation is fit to a dataset with  equal to 50.  The confidence interval for the predicted value of y when x is equal to 40 is tighter than the confidence interval for the predicted value of y when x is equal to 65. Assume that 40 and 65 were both in the original sample that generates the regression equation.

2.(a)Nine people are going on a skiing trip in 3 cars that will hold 2, 4, and 5 persons, respectively.  In how many ways is it possible to transport the 9 people to the ski lodge using all cars?  Exact numerical answer required.

(b)Three students work independently on a homework problem.  Each student has a 75% chance of solving the problem correctly. The students then compare the final answer to check their work. What is the probability that exactly one student gets the correct answer?  What is the probability that at least one student gets the correct answer?  Exact numerical answer required.

(c)Three balls are drawn randomly out a bag without replacement.  The bag initially contains seven orange balls, eight green balls, and two white balls.  What is the probability of picking at least one orange ball and at least one white ball? Exact numerical answer required

2. Probability

(d)Twenty five equally qualified people apply for five job openings.  Thirteen of the applicants come from city A and 12 of the applicants come from city B.  Assuming the selection process is unbiased, what is the probability that four of the people who are hired come from city B?  Exact numerical answer required.

3.The salt content of a prepared food is measured in a random sample of n=10 yielding a sample standard deviation of s=4.8 (milligrams). 

(a)Test the hypothesis H0: σ2 = 18; H1: σ2 ≠ 18 using a p-value with α = 0.05.

(b)Suppose the actual standard deviation is twice as large as the value listed in H0.  What is the power of the test?

(c)Suppose that the true variance is σ2 = 40.  How large would the sample size need to be in order to achieve a power of at least 0.9?

4.The concentration of active ingredient in a liquid laundry detergent is thought to be affected by the type of catalyst used in the process.  Five observations on concentration are taken with each catalyst, yielding the following data: 

Catalyst 1: 57.9, 66.2, 65.4, 65.2, 63.7

Catalyst 2: 66.4, 71.7, 70.3, 69.3, 69.6

(a)Find a 95% confidence interval on the difference in mean active concentrations for the two catalysts. Assume σ1=σ2. 

(b)Using the results from part (a), is there any evidence to indicate that the mean active concentrations depend on the choice of catalyst?  Assume σ1=σ2. 

(c)Repeat question (b) but assume σ1≠σ2.  

(d)Suppose the true difference in active concentration is 5 grams per liter.  What is the power of the test to detect this difference if α = 0.05?  

(e)What sample size would be needed to achieve a power of 90% for the conditions listed in part (d)?

5.The arsenic concentration in drinking water is measured in two different communities.  The samples may be summarized with the information n1=10, n2=10, .  Assume that both populations are normal.

(a)Test the hypothesis that σ12 = σ22.  Use a significance level of α=0.05. 

(b)Formulate a hypothesis test and perform necessary calculations to answer the question: is the arsenic concentration higher in the drinking water of community two than community one?  Use a significance level of α=0.05

(c)Estimate the equal sample size required to detect a difference in the true means of the arsenic concentrations as small as 10 ppb with a power of 90%. 

6. A study of 250 subjects considered the relationship between age (x) and weight (y).  Summary statistics for each variable are shown below:  

(a)Find the least squares prediction equation (i.e. ).  

(b)Conduct the hypothesis test to determine whether the estimated slope is significant using a p-value.  Use α = 0.05.  

(c)Conduct the hypothesis test to determine whether the estimated intercept is significant using a p-value.  Use α = 0.05.  

(d)What is R2 for this equation?  And explain its meaning in words.