B.38  

Average Size of a Performing Group in the Rock and Roll Hall of Fame

From its founding through 2015, the Rock and Roll Hall of Fame has inducted 303 groups or individuals, and 206 of the inductees have been performers while the rest have been related to the world of music in some way other than as a performer. The full dataset is available at RockandRoll on StatKey. Some of the 206 performer inductees have been solo artists while some are groups with a large number of members. We are interested in the average number of members across all groups or individuals inducted as performers.

(a)  

What is the mean size of the performer inductee groups (including individuals)? Use the correct notation with your answer.

(b)  

Use technology to create a graph of all 206 values. Describe the shape, and identify the two groups with the largest number of people.

(c)  

Use technology to generate a sampling distribution for the mean size of the group using samples of size n = 10. Give the shape and center of the sampling distribution and give the standard error.

(d)  

What does one dot on the sampling distribution represent?

If the moment-generating function of X is M(t) = exp(3 t + 12.5 t²) = e^{3 t + 12.5 t2}.

a. Find the mean and the standard deviation of X.

Mean =

standard deviation =

b. Find P(4 < X < 16). Round your answer to 3 decimal places.

c. Find P(4 < X² < 16). Round your answer to 3 decimal places.

11.A computer manufacturer estimates that its cheapest screens will last less than 2.8 years. A random sample of 61 of these screens has a mean life of 2.5 years. The population is normally distributed with a population standard deviation of 0.88 years. At α=0.02, what type of test is this and can you support the organization’s claim using the test statistic?

Claim is alternative, reject the null and support claim as test statistic (-2.66) is in the rejection region defined by the the critical value (-2.05)

Claim is alternative, fail to reject the null and cannot support claim as test statistic (-2.66) is in the rejection region defined by the critical value (-2.05)

Claim is null, reject the null and support claim as test statistic (-2.66) is in the rejection region defined by the critical value (-2.05)

Claim is null, fail to reject the null and cannot support claim as test statistic (-2.66) is in the rejection region defined by the critical value (-2.05)

12. A pharmaceutical company claims that the average cold lasts an average of 8.4 days. They are using this as a basis to test new medicines designed to shorten the length of colds. A random sample of 106 people with colds, finds that on average their colds last 8.28 days. The population is normally distributed with a standard deviation of 0.9 days. At α=0.02, what type of test is this and can you support the company’s claim using the p-value?

Claim is null, reject the null and cannot support claim as the p-value (0.170) is less than alpha (0.02)

Claim is null, fail to reject the null and support claim as the p-value (0.170) is greater than alpha (0.02)

Claim is alternative, fail to reject the null and support claim as the p-value (0.085) is less than alpha (0.02)

Claim is alternative, reject the null and cannot support claim as the p-value (0.085) is greater than alpha (0.02)

13. A business receives supplies of copper tubing where the supplier has said that the average length is 26.70 inches so that they will fit into the business’ machines. A random sample of 48 copper tubes finds they have an average length of 26.77 inches. The population standard deviation is assumed to be 0.20 inches. At α=0.05, should the business reject the supplier’s claim?

No, since p>α, we reject the null and the null is the claim

No, since p>α, we fail to reject the null and the null is the claim

Yes, since p>α, we fail to reject the null and the null is the claim

Yes, since p<α, we reject the null and the null is the claim

14. The company’s cleaning service states that they spend more than 46 minutes each time the cleaning service is there. The company times the length of 37 randomly selected cleaning visits and finds the average is 47.2 minutes. Assuming a population standard deviation of 5.2 minutes, can the company support the cleaning service’s claim at α=0.10?

Yes, since p>α, we reject the null. The claim is the null, so the claim is not supported

Yes, since p<α, we fail to reject the null. The claim is the null, so the claim is not supported

No, since p>α, we fail to reject the null. The claim is the alternative, so the claim is not supported

No, since p<α, we reject the null. The claim is the alternative, so the claim is supported

15.. A customer service phone line claims that the wait times before a call is answered by a service representative is less than 3.3 minutes. In a random sample of 62 calls, the average wait time before a representative answers is 3.26 minutes. The population standard deviation is assumed to be 0.29 minutes. Can the claim be supported at α=0.08?

No, since test statistic is not in the rejection region defined by the critical value, reject the null. The claim is the alternative, so the claim is not supported

Yes, since test statistic is in the rejection region defined by the critical value, reject the null. The claim is the alternative, so the claim is supported

Yes, since test statistic is in the rejection region defined by the critical value, reject the null. The claim is the alternative, so the claim is supported

No, since test statistic is not in the rejection region defined by the critical value, fail to reject the null. The claim is the alternative, so the claim is not supported

3.For variables measured at the nominal level, what values can the measures of association take on? What about variables at the ordinal and interval/ratio levels?

Does pollution increase mean death rate? A researcher sampled 31 “unpolluted” areas greater than 50 km away from industrial plants, and 23 different “polluted” areas near industrial plants. The average mortalities in the unpolluted areas were 3 deaths per day per 100000 people (with a sample standard deviation of 0.4 deaths/day/100000 people), and was 3.3 deaths per day per 100000 people (with a sample standard deviation of 0.5 deaths/day/100000 people) in the polluted area. At the alpha=0.01 level, answer the question does pollution increase average death rate? Show statistical and critical values as appropriate. Assume that variances are equal.

  You are studying the relationship between smoking and hair loss. You find a positive moderate effect size. You conclude:

	A.	there is clinical significance
	B.	the correlation between smoking and hair loss is between -0.3 and -0.5
	C.	the correlation between smoking and hair loss is between 0.3 and 0.5
	D.	this is a significant relationship

A tax auditor is selecting a sample of 5 tax returns for an audit. If 2 or more of these returns are​"improper," the entire population of 50 tax returns will be audited. Complete parts​ (a) through​ (e) below.

Q. What is the probability that the entire population will be audited if the true number of improper returns in the population is:

a) 15

b) 20

c) 5

d) 10

Review these Skill Builders (and all of the other Course Materials): -Evaluating p-values -Statistical Power Identify the scenario you are evaluating and name the population. Estimate the size of that population. Example: the population of scenario 1 seems to be students at a State University so you could estimate the number of students at a typical State University. The University of South Florida up the road from me has about 40,000 students Identify the independent variable (IV) and the dependent variable (DV). Sometimes this is stated by the researchers and sometimes you have to ferret it out. In scenario 2, the IV and DV are given as Race and Education, respectively. Write a null hypothesis. If the null hypothesis is not provided in the scenario, write a null hypothesis based on the information that is provided in the scenario. Each scenario addresses differences in an interval or ratio DV among a Nominal or Ordinal IV made up of 2 or more groups. So write the null hypothesis this way: There is no difference in Education based on Race among (state/name the population). Critically evaluate the sample size. This is tricky because the scenarios do not provide us with the right information to calculate an appropriate sample size. And you want to avoid stating that a sample size ‘seems’ to be the right size (very amateurish). What to do? Go to this sample size calculator: https://www.surveysystem.com/sscalc.htm. Use the box labeled Calculate Sample Size, Enter .95 for the confidence level, your estimate of the population, 5 for the confidence interval and see what pops up for the ideal sample size. Compare that number to the sample size in the scenario and critically evaluate the sample size in terms of making a Type I or Type II error. For example, if the sample size is smaller than the ideal sample size, does the probability of making a Type I error increase or decrease. Do the same drill with a Type II error. Critically evaluate the scenario for meaningfulness. Follow the guidance I provided in the Announcement Week 5 Discussion: How To Critically Evaluate The Discussion Scenario. Note: we can often relate meaningfulness to social change. That is, if the research is meaningful then it may have implications for social change. Try evaluating meaningfulness and social change in the same paragraph. But first, define meaningfulness and define social change. Cite, cite, cite. Critically evaluate the statements for statistical significance. Compare the researcher reported p-value for the hypothesis test they conducted (either a t-test or an ANOVA) to the confidence level (usually .05). If the reported p-value is greater than .05, then the researcher should fail to reject the null hypothesis and state that there is no statistical significance. If the reported p-value is less than .05, then the researcher should reject the null hypothesis and state that there is statistical significance. I know this is counter-intuitive. Just do it. Add this for grins, “There is no such decision as ‘rapidly approaching significance.’ This is statistics, not a hurricane watch.” Select 1 response to the following multiple choice question: What scenario would you find to be the least fun?Having a root canal performed by an experienced dentist. Having 4 root canals performed by an unsupervised novice dentist. Having 21 root canals performed by a trained Capuchin monkey. Trying to statistically determine differences in patient post-root canal pain levels based on the dentist’s training.

A researcher wants to test whether the mean lengths of two species of trout are the same. He obtains the weights of 10 individuals of species A and 10 individuals of species B. The sample mean weights are 12.3 kg and 14.3 kg, respectively, and the sample variances are 3.5 kg² and 4.5kg². Assume that the lengths are normally distributed.  

a. Should we retain the hypothesis that the population variances are equal?

b. Should we retain the hypothesis that the population means are equal?

The following regression output was obtained from a study of architectural firms. The dependent variable is the total amount of fees in millions of dollars.

Predictor	Coef	SE Coef	T	P
Constant	7.987	2.967	2.69	-
X1	0.12242	0.03121	3.92	0.0000
X2	-0.12166	0.05353	-2.27	0.028
X3	-0.06281	0.03901	-1.61	0.114
X4	0.5235	0.1420	3.69	0.001
X5	-0.06472	0.03999	-1.62	0.112
Analysis of Variance
Source	DF	SS	MS	F	P
Regression	5	3710.00	742.00	12.89	0.000
Residual Error	46	2647.38	57.55
Total	51	6357.38

X1 - # of architects employed by the company

X2 - # of engineers employed by the company

X3 - # of years involved with health care projects

X4 - # of states in which the firm operates

X5 - % of the firms work that is health care-related

1. Write out the regression equation
2. How large is the sample? How many independent variables are there?
3. Conduct a global test of hypothesis to see if any of the set of regression coefficients could be different from 0. Use the .05 significance level. What is your conclusion?
4. Conduct a test of hypothesis for each independent variable. Use the .05 significance level. Which variable would you consider eliminating first?
5. Outline a strategy for deleting independent variables in this cas

Use R to complete the following questions. You should include your R code, output and plots in your answer.

1. Two methods of generating a standard normal random variable are:

a. Take the sum of 5 uniform (0,1) random numbers and scale to have mean 0 and standard deviation 1. (Use the properties of the uniform distribution to determine the required transformation).

b. Generate a standard uniform and then apply inverse cdf function to obtain a normal random variate (Hint: use qnorm).

For each method generate 10,000 random numbers and check the distribution using

a. Normal probability plot

b. Mean and standard deviation

c. The proportion of the data lying within the theoretical 2.5 and 97.5 percentiles and the 0.5 and 99.5 percentiles. (Hint: The ifelse function will be useful)  

If the probability that a family will buy a vacation home in Manmi, Malibu, or Newport is 0.25, 0.10, 0.35, what is the probability the family will consummate one of these transactions? please show all wor with explanation.

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities.

What are the chances that a person who is murdered actually knew the murderer? The answer to this question explains why a lot of police detective work begins with relatives and friends of the victim! About 68% of people who are murdered actually knew the person who committed the murder.† Suppose that a detective file in New Orleans has 65 current unsolved murders. Find the following probabilities. (Round your answers to four decimal places.)

(a) at least 35 of the victims knew their murderers

(b) at most 48 of the victims knew their murderers

(c) fewer than 30 victims did not know their murderers

(d) more than 20 victims did not know their murderers

Mr. James McWhinney, president of Daniel-James Financial Services, believes there is a relationship between the number of client contacts and the dollar amount of sales. To document this assertion, Mr. McWhinney gathered the following sample information. The X column indicates the number of client contacts last month, and the Y column shows the value of sales ($ thousands) last month for each client sampled.

Number of Contacts (X)	Sales ($ thousands) (Y)
14	24
12	14
20	28
16	30
46	80
23	30
48	90
50	85
55	120
50	110

1. Determine the standard error of estimate
2. Determine the coefficient of determination. Interpret the coefficient of determination

Professor Nord stated that the mean score on the final exam from all the years he has been teaching is a 79%. Colby was in his most recent class, and his class’s mean score on the final exam was 82%. Colby decided to run a hypothesis test to determine if the mean score of his class was significantly greater than the mean score of the population. α = .01.  If p = 0.29

What is the mean score of the population? What is the mean score of the sample? What should Colby’s statement of conclusion be