Question 1: We want to estimate the mean change in score µ in the population of all high school seniors. An SRS of 450 high school seniors gained an average of  x⎯⎯⎯x¯ = 21 points in their second attempt at the SAT Mathematics exam. Assume that the change in score has a Normal distribution with standard deviation 52.201.

Find σx¯, the standard deviation of the mean change x¯ _______ (±±0.001).

Using the 68-95-99.7 Rule (Empirical Rule), give a 95% confidence interval for μμ based on this sample.

Confidence interval (±±0.001) is between _____ and _______.

Question 3: We have the survey data on the body mass index (BMI) of 670 young women. The mean BMI in the sample was x¯=25.3. We treated these data as an SRS from a Normally distributed population with a standard deviation σ=7.8.

Give confidence intervals for the mean BMI and the margins of error for 90%, 95%, and 99% confidence.

Recent crime reports indicate that 17.3 motor vehicle thefts occur every hour in Canada. Assume that the distribution of thefts per hour can be approximated by a Poisson probability distribution.

a. Calculate the probability exactly four thefts occur in an hour.(Round the final answer to 5 decimal places.)

Probability            

b. What is the probability there are no thefts in an hour? (Round the final answer to 5 decimal places.)

Probability            

c. What is the probability there are at least 20 thefts in an hour? Use excel or online calculator to find the answer. (Round the final answer to 5 decimal places.)

Probability            

probabilities Find the probabilities for each event. Consider rolling a pair of fair dice two times. Let A be the total on the up-faces for the first roll and let B be the total on the up-faces for the second roll.

1. A = {2 on the first roll}, B = {8 or more on the first roll}.
2. A = {2 on the first roll}, B = {8 or more on the second roll}.
3. A = {5 or less on the second roll}, B = {4 or less on the first roll}.
4. A = {5 or less on the second roll}, B = {4 or less on the second roll}.

Everyday before work Sally feeds her cat either tuna or chicken cat food. If she feeds her cat tuna today, then tomorrow she will roll a 6-sided die. If the roll is a 5 or a 6 then she will feet her cat tuna again tomorrow otherwise she will feed her cat chicken. If she feed her cat chicken cat food today, she will feed her cat tuna tomorrow and not chicken. On the first day, Sally flips a coin to decide what to feed her cat. If its heads - its tuna and if she flips tails its chicken.

Name the states for this Markov chain and then draw a transition diagram.

Provide a transition matrix P, and the initial state distribution So for this Markov chain.

give two examples in any area of interest to you (other than those already presented in this chapter) where regression analysis can be used as a data analytic tool to answer some questions of interest. For each example:

a. what is the question of interest?

b. Identify the response and the predictor variable.

c. Classify each of the variables as either quantitative or qualitative.

d. Which type of regression (see table 1.15) can be used to analyze the data?

e. Give a possible form of the model and identify its parameters.

Table 1.15

Type of Regression

Univariate. Multivaritae - Only one quantiative response variable. Two or more quantitative response variables

Simple. Mutiple - Only one predictor variable. Two or more predictor variables

Linear - All parameteres enter the equation linearly, possibly after transformaton of the data

Nonlinear - The relationship between the response and some of the predictors is nonlinear or some of the parameteres appear nonlinearly, but no transformation is possible to make the parameters appear linearly.

Analysis of variance - All predictors are qualitative variables

Analysis of covariance - Some predictors are quantitative variables and others are qualitative variables

Logistic - The response Variables is qualitative

1. The concentration of copper in the sediments in a particular stream in the Yukon was studied using 40 samples. The mean of the samples was found to be 40.0 ppm, with a standard deviation of 16.0 ppm.

a) Construct a 95% confidence interval for the true mean level of copper in the stream sediments.

              b) Suppose that 70 samples were collected (instead of 40), would the 95% confidence interval widen or tighten? Explain how you know.

              c) Suppose that 40 samples were collected you wanted to be 99% confident (instead of 95%), would the confidence interval widen or tighten compared to a)? Explain.

              d) Suppose that 20 samples were collected; what additional piece of information would you need in order to construct a 95% confidence interval?

The admissions officer at a small college compares the scores on the Scholastic Aptitude Test (SAT) for the school's in-state and out-of-state applicants. A random sample of 8 in-state applicants results in a SAT scoring mean of 1048 with a standard deviation of 44. A random sample of 1616 out-of-state applicants results in a SAT scoring mean of 1147 with a standard deviation of 43. Using this data, find the 98% confidence interval for the true mean difference between the scoring mean for in-state applicants and out-of-state applicants. Assume that the population variances are not equal and that the two populations are normally distributed.

Step 2 of 3 : Find the margin of error to be used in constructing the confidence interval. Round your answer to six decimal places.

Find, or come up with, a data set to test the equality of means of 3 categories. Provide the sample statistics for each category. Using technology (Ti-84 or Excel) find the critical value, test statistic and p-value of the ANOVA test. Then interpret the results in the context of the problem.

You are the director of a cardiac surgical unit and you are interested in the difference between surgical times for experienced surgeons versus newly-trained surgeons. You collect data on a random selection of surgical operating room time (in minutes) for triple bypass surgery for 12 experienced surgeons and 14 newly-trained surgeons. The patients for all 26 surgeries were similar across a range of clinical and demographic characteristics. Use these data to answer the questions below. Show all supporting calculations.

1. Assuming the distributions of surgical times for both types of surgeons are normally-distributed with known standard deviations of 80 minutes for experienced surgeons and 40 minutes for newly-trained surgeons, conduct the appropriate statistical test to determine if the mean surgical times differ between the two types of surgeons. Assume α=0.05.  

2. Subsequently you learned that the surgical operating room times for the newly-trained surgeons were incorrectly recorded and that each of the values were 30 minutes too short. What, if any, difference does this make to the conclusion you reached in question a?

You are a nurse manager working at an outpatient stroke rehabilitation clinic in Hamilton, Ontario. You are interested in whether client wait times at your clinic (from arrival time to the time the client is served by the health care professional) differ from those at another outpatient stroke rehabilitation clinic in Oakville, Ontario. The table below provides the client wait times that you have obtained for a random sample of clients from both clinics. You know from previous studies that wait times are approximately normally distributed with equal variance. Use these data to answer the questions below. Assume α=0.05. Show all supporting calculations.

1. Can you conclude that the wait times at the two clinics are equal?

2. What is the 95% confidence interval for the mean difference in wait times for the two clinics?

3. Does the confidence interval from question b support the conclusion you reached in question a. Explain.

You are a nurse working in a cardiac unit of the hospital. You are interested in whether blood pressure readings differ depending on the patient’s position. You have collected the systolic blood pressure (SBP) readings for 12 patients in two different positions (supine, standing). The table below provides the results. Assume that SBP differences are approximately normally distributed. Conduct the appropriate test to determine whether SBP differs for the two positions. Assume α=0.05 and show all supporting calculations.

You are a public health nurse working in an elementary school in Hamilton, Ontario. You have seen a substantial increase in the number of children in your school presenting with Attention Deficit Disorder (ADD). You are interested in whether a new drug developed by a local drug company improves the ability of children with ADD to maintain attention. You obtain ethics approval to study the drug, informed consent from the parents of 24 fifth grade children with ADD, administer the drug to the 24 children, and administer a test to the children to determine their ability to maintain attention while performing a task. Scores are continuous, can range from 0 to 25 with higher scores reflecting a better ability to maintain attention, and you know from previous research that scores tend to be normally distributed. Six of these students receive a placebo containing none of the drug. Six students receive 2 mg of the drug, six students receive 4 mg of the drug, and six students receive 6 mg of the drug. The table below summarizes the results from your study. Use these data to answer the following questions. Assume α=0.05.

1. 1. You hypothesize that there will be a difference among the groups in the ability to maintain attention. Run the appropriate statistical test to determine if your hypothesis is correct. Was your hypothesis confirmed by the study results? Provide the calculation and/or SPSS output to support your answer.  
2. 2. You have one further prediction, and that is that children with ADD who receive the drug will perform significantly better than children with ADD who do not receive the drug. Conduct the appropriate test to confirm this prediction, if the results from question a support doing this additional testing. Are you able to confirm your prediction? Explain, including the calculations and/or SPSS output to support your answer.

You are a nurse working for the Public Health Unit in Hamilton, Ontario. You know that each year many people seek treatment from their family doctors for colds, and that many people need to take time off work to recover from them. A drug company has asked you to assist with a study to test a new drug that they have developed to prevent colds. You recruit 100 women and 200 men from a population of 100,000 people that volunteer to be in the study. At the end of the study, you find that 38% of women caught a cold and 51% of men caught a cold. Use these study data to answer the following questions. Show all supporting calculations.

1. It would appear from comparing the proportion of women and men that caught a cold, that the drug may be more effective in women. Can you conclude this? Assume α=0.05.

2. Can you conclude the drug is more effective in women than men if you assume α=0.01?

3. The drug company claims that the drug is equally effective for men and women. Can you reject the company’s claim? Use the same statistical procedure and level of significance that you used in question a to answer this question.   

4. What is the p-value for the statistical test you conducted in question c?

5. What is an alternative statistical procedure that could be used to answer question c? What would the test statistic and associated p-value be for this statistical procedure?