A researcher would like to evaluate the effectiveness of a pain-relief patch designed for lower back pain. Prior to testing the patch, each of n = 4 patients rates the current level of back pain on a scale from 1 to 8. After wearing the patch for 90 minutes, a second pain rating is recorded. The data are as follows.

​

                                                Before        After

                                          __________________

                   G = 36                  6          2

                  G² = 784                6          2

                ΣX² = 192                4          4

                                                8          4

                                         ______________________

                                            T = 24   T = 12

                                          SS = 8    SS = 4

Perform an ANOVA and then analyze the data with a t-test. Use the .05 level of significance for both.

An independently selected sample of five men also participated in the same study. The table below shows results for the number of pounds lost by the five men and the five women in the study. The researcher will use the .01 significance level to test whether (on average) the program produces different weight loss results for men and women. You may assume the population variances are equal (although the sample variances are not).

f. Formulate the hypothesis for this test.

g. Should the pooled-sample variance be used in this situation? Why?

h. Choose the appropriate formula for the test statistic and find its value.

i. What is the rejection region for this test?

j. What should the researcher conclude?

St. Andrew’s University receives 900 applications annually from prospective students. The application forms contain a variety of information including the individual’s scholastic aptitude test (SAT) score and whether or not the individual desires on-campus

housing.

What is the probability that a simple random sample of 30 applicants will provide an estimate of the population mean SAT score that is within plus or minus 10 (within 10 points) of the actual population mean m ? Given that the population Standard Deviation is 80.

Data show 648 applicants wanting On-Campus Housing. What is the probability that sample proportion exceeds 50%, when n =30?

What is the probability that a simple random sample of 30 applicants will provide an estimate of the population proportion of applicants desiring on-campus housing that is within plus or minus .05 of the actual population proportion?

If the University can provide to no more than 45% for the on –campus housing facilities, what would be the estimated number of accepted applicants desiring on-campus housing?

4. Suppose a company wants to find the determinants of job satisfaction for its employees. The company gathers some anonymous data from its employees and runs a regression with Job Satisfaction (on a 1 to 10 scale) as the response variable, and it gets the following results.

ANOVA

(a) What are the null and alternative hypotheses (regarding the b values) for the F-test for the regression?

(b) What is the F-statistic for this regression

(c) The critical F* for the test is 5.04. Do we reject or not reject the null hypothesis in this case?

Why?

(d) What percentage of the variation in Job Satisfaction is “explained” by this regression?

Total plasma volume is important in determining the required plasma component in blood replacement therapy for a person undergoing surgery. Plasma volume is influenced by the overall health and physical activity of an individual. Suppose that a random sample of 50 male firefighters are tested and that they have a plasma volume sample mean of x = 37.5 ml/kg (milliliters plasma per kilogram body weight). Assume that σ = 7.40 ml/kg for the distribution of blood plasma.

(a) Find a 99% confidence interval for the population mean blood plasma volume in male firefighters. What is the margin of error? (Round your answers to two decimal places.)

(b) What conditions are necessary for your calculations? (Select all that apply.)

n is large

σ is known

the distribution of weights is normal

σ is unknown

the distribution of weights is uniform

(c) Interpret your results in the context of this problem.

1% of the intervals created using this method will contain the true average blood plasma volume in male firefighters.

99% of the intervals created using this method will contain the true average blood plasma volume in male firefighters.    

The probability that this interval contains the true average blood plasma volume in male firefighters is 0.01.

The probability that this interval contains the true average blood plasma volume in male firefighters is 0.99.

(d) Find the sample size necessary for a 99% confidence level with maximal margin of error E = 2.60 for the mean plasma volume in male firefighters. (Round up to the nearest whole number.)
  male firefighters

There are many controversies in the real life about the hypothesis testing. What do you think are the potential reasons for that? Can we find and share any example of misleading results of the hypothesis testing?

Adam, Brian, Chris and Donald are taking goal shots in soccer. They take shots in the order, Adam, Brian, Chris, Donald, then repeat. By going in this order they have an equal chance of making a goal first. Donald makes a goal 50% of the time. What would be the probability of Adam winning if he went last instead of first?

When σ is unknown and the sample is of size n ≥ 30, there are two methods for computing confidence intervals for μ.

Method 1: Use the Student's t distribution with d.f. = n − 1.
This is the method used in the text. It is widely employed in statistical studies. Also, most statistical software packages use this method.

Method 2: When n ≥ 30, use the sample standard deviation s as an estimate for σ, and then use the standard normal distribution.
This method is based on the fact that for large samples, s is a fairly good approximation for σ. Also, for large n, the critical values for the Student's t distribution approach those of the standard normal distribution.

Consider a random sample of size n = 31, with sample mean x = 44.7 and sample standard deviation s = 5.4.

(a) Compute 90%, 95%, and 99% confidence intervals for μ using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

(b) Compute 90%, 95%, and 99% confidence intervals for μ using Method 2 with the standard normal distribution. Use s as an estimate for σ. Round endpoints to two digits after the decimal.

(c) Compare intervals for the two methods. Would you say that confidence intervals using a Student's t distribution are more conservative in the sense that they tend to be longer than intervals based on the standard normal distribution?

Yes. The respective intervals based on the t distribution are longer.

No. The respective intervals based on the t distribution are longer.    

Yes. The respective intervals based on the t distribution are shorter.

No. The respective intervals based on the t distribution are shorter.

(d) Now consider a sample size of 71. Compute 90%, 95%, and 99% confidence intervals for μ using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

(e) Compute 90%, 95%, and 99% confidence intervals for μ using Method 2 with the standard normal distribution. Use s as an estimate for σ. Round endpoints to two digits after the decimal.

(f) Compare intervals for the two methods. Would you say that confidence intervals using a Student's t distribution are more conservative in the sense that they tend to be longer than intervals based on the standard normal distribution?

Yes. The respective intervals based on the t distribution are shorter.

No. The respective intervals based on the t distribution are longer.    

No. The respective intervals based on the t distribution are shorter.

Yes. The respective intervals based on the t distribution are longer.

With increased sample size, do the two methods give respective confidence intervals that are more similar?

As the sample size increases, the difference between the two methods remains constant.

As the sample size increases, the difference between the two methods becomes greater.    

As the sample size increases, the difference between the two methods is less pronounced.

State H0 and H1. Answer the question. Use megastat and submit software output.

See Worksheet 4. John Isaac Inc., a designer and installer of industrial signs, employs 60 people. The company recorded the type of the most recent visit to a doctor by each employee. A national assessment conducted in 2014 found that 53% of all physician visits were to primary care physicians, 19% to medical specialists, 17% to surgical specialists, and 11% to emergency departments. Test at the .01 significance level if Isaac employees differ significantly from the national survey distribution.

The home run percentage is the number of home runs per 100 times at bat. A random sample of 43 professional baseball players gave the following data for home run percentages.

(a) Use a calculator with mean and standard deviation keys to find x and s. (Round your answers to two decimal places.)

(b) Compute a 90% confidence interval for the population mean μ of home run percentages for all professional baseball players. Hint: If you use the Student's t distribution table, be sure to use the closest d.f. that is smaller. (Round your answers to two decimal places.)

(c) Compute a 99% confidence interval for the population mean μ of home run percentages for all professional baseball players. (Round your answers to two decimal places.)

(d) The home run percentages for three professional players are below.

Examine your confidence intervals and describe how the home run percentages for these players compare to the population average.

We can say Player A falls close to the average, Player B is above average, and Player C is below average.

We can say Player A falls close to the average, Player B is below average, and Player C is above average.    

We can say Player A and Player B fall close to the average, while Player C is above average.

We can say Player A and Player B fall close to the average, while Player C is below average.

(e) In previous problems, we assumed the x distribution was normal or approximately normal. Do we need to make such an assumption in this problem? Why or why not? Hint: Use the central limit theorem.

Yes. According to the central limit theorem, when n ≥ 30, the x distribution is approximately normal.

Yes. According to the central limit theorem, when n ≤ 30, the x distribution is approximately normal.    

No. According to the central limit theorem, when n ≥ 30, the x distribution is approximately normal.

No. According to the central limit theorem, when n ≤ 30, the x distribution is approximately normal.

1. Plz do all the calculations on excel n show the excel files.
2. Construct and interpret a correlation matrix for your data.
3. Show the step wise process of determining the best regression model to predict PRICE Anova. EXPLAIN the process as you move from the full model to your final model.
4. Using your final model, select values for the independent variables and predict the house’s sales price.

Listed below are the log body weights and log brain weights of the primates species in the data set ”mammals”. Find the equation of the least squares line with y = log brain weight and x = log body weight. Do it by hand, by constructing a table like the one in Example 9.1. Then do it with your calculator as efficiently as possible. Finally, use the lm function in R to do it by creating a linear model object ”primates.lm”. The model formula is ”log(brain)∼log(body)”. You can select the primates and put them in a new data frame by first listing the primate species names:

> primatenames=c(”Owl monkey”, ”Patas monkey”, ”Gorilla”, etc.)

and then

> primates=mammals[primatenames, ]

Your ”data” argument in calling lm would be ”data=primates”, as in

> primates.lm=lm(log(brain)∼log(body),data=primates)

Alternatively, you can just use ”mammals[primatenames, ]” as the data argument in lm, that is,

> primates.lm=lm(log(brain)∼log(body), data=mammals[primatenames,])

log body               log brain

Owl monkey                       -0.7339692         2.740840

Patas monkey                    2.3025851           4.744932

Gorilla                                   5.3327188           6.006353

Human                                 4.1271344           7.185387

Rhesus monkey                                1.9169226           5.187386

Chimpanzee                                       3.9543159           6.086775

Baboon                                                 2.3561259           5.190175

Verbet                                                  1.4327007           4.060443

Galago                                                  -1.6094379        1.609438

Slow loris                                             0.3364722           2.525729

Let x be a random variable that represents red blood cell (RBC) count in millions of cells per cubic millimeter of whole blood. Then x has a distribution that is approximately normal. For the population of healthy female adults, the mean of the x distribution is about 4.8 (based on information from Diagnostic Tests with Nursing Implications, Springhouse Corporation). Suppose that a female patient has taken six laboratory blood tests over the past several months and that the RBC count data sent to the patient’s doctor are 4.9, 4.2, 4.5, 4.1, 4.4, 4.3

(a) Use a calculator to verify that ?̅= 4.40 and ? = 0.28.

(b) Do the given data indicate that the population mean RBC count for this patient is lower than 4.8? Use ? = .05.

(c) Obtain a 90% confidence interval for the mean RBC count.

(d) Can you come to the same conclusion you made in (b) using the interval approach in (c).

What is the impact of widening the margin of error?

Question 9 options:

The management at an environmental consulting firm claims the mean weekly salary is $275 with a standard deviation of $34.10.

1. If we want to address the question “If management’s claims are true, what is the probability that an individual worker would make an average weekly salary less than $264.50?”, write the probability statement.

2. If we want to address the question “If management’s claims are true, what is the probability that an individual worker would make an average weekly salary less than $264.50?”, what is the probability? Be sure to show how you calculated your probability.

3. If we want to address the question “If management’s claims are true, what is the probability that a group of 27 workers would make an average weekly salary less than $264.50?”, write the probability statement.

4. If we want to address the question “If management’s claims are true, what is the probability that a group of 27 workers would make an average weekly salary less than $264.50?”, what is the probability? Be sure to show how you calculated your probability.