In your attempt to “catch them all” you hunt the prized Pokemon, Mew and Mewtwo. As you battle them with your top 25 Pokemon, you lose every battle. However, after each battle, you notice that the hit points (HP) of Mew and Mewtwo have been reduced. The remaining HPs of Mew and Mewtwo are shown below after battling your respective Pokemon. (A higher HP remaining indicates more success in battle.) You suspect that Mewtwo is the tougher opponent, and you want to test this hypothesis at the 95% level of confidence.

Assume that before each battle, Mew and Mewtwo started with the same number of HPs.

a. State the null hypotheses for this test.

b. State the alternative hypotheses for this test.

c. What type of test will you use to test this hypothesis?

d. Describe the Type I Error for this test, and explain its implication for you as you continue your quest.

e. Describe the Type II Error for this test, and explain its implication for you as you continue your quest.

f. What is the rejection region for this hypothesis test?

g. Calculate the test statistic for this hypothesis test. What is your conclusion?

h. If you reject the null hypothesis for this test, what is the probability that you have committed a Type I Error? (i.e., find the p-value for this test.)

i. What action will you take as you continue your quest based on your conclusion.

A poll found that of 79 registered voters under 45 years of age, 18 favored a particular candidate. Of 65 registered voters aged at least 45 years old, 15 favored the same candidate.

Using α=0.1α=0.1, test the claim that the two age groups support this candidate in equal proportions

In a study of smoking and its effects on sleep patterns, one variable is the time that it takes to fall asleep. A random sample of size 12 is drawn from the population of smokers; an independent sample of size 15 is drawn from the population of non-smokers. The mean time to sleep for smokers is 43.70 minutes; for non-smokers it is 30.32 minutes. The sample variance for the time to sleep for smokers is 286.549 min2  while for non-smokers it is 50.806 min2 .

Find a 95% confidence interval to determine if these data indicate that smokers tend to take longer to fall asleep than non-smokers. (Hint: Degrees of freedom formula yields df = 14.2)

Given a random sample of size 16 from a normally distributed population with a mean of 100 and standard deviation of 24, find P(x bar≤90)

The 2011 National Health Interview Survey by the National Center for Health Statistics (NCHS) provides weight categorization for adults 1818 years and older based on their body mass index.

The same NCHS report breaks down the sampled individuals by age group. The percentages of obese individuals in the 2011 survey for each age group are provided in the table.

What do you conclude about the weight problem in the United States?

A: Only adults in the age group 4545 to 6464 appear to have obese individuals.

B: All age groups have a large rate of obesity. Adults 1818 to 4444 and 6565 to 7474 have the highest percentage of obese individuals.

C: All age groups have a relatively small rate of obesity. The weight problem in the United States appears to have been exaggerated.

D: All age groups have a large rate of obesity. Adults 4545 to 6464 and 6565 to 7474 have the highest percentage of obese individuals.

(b) Could you make a single pie chart for these data?

A: Yes, because the percentages in the four age groups do not add up to 100%100% .

B: No, because the percentages in the four age groups add up to 100%100% .

C: No, because the percentages in the four age groups do not add up to 100%100% .

D: Yes, because the percentages in the four age groups add up to 100%100% .

The following three independent random samples are obtained from three normally distributed populations with equal variance. The dependent variable is starting hourly wage, and the groups are the types of position (internship, co-op, work study).

Do not forget to convert this table from parallel format (i.e., groups in each column) to serial format for analysis in SPSS.

Use SPSS (or another statistical software package) to conduct a one-factor ANOVA to determine if the group means are equal using α=0.05α=0.05. Though not specifically assessed here, you are encouraged to also test the assumptions, plot the group means, and interpret the results.

Group means (report to 2 decimal places):
Group 1: Internship: ____
Group 2: Co-op: ____  
Group 3: Work Study: ____  


ANOVA summary statistics:
F-ratio = ____
(report accurate to 3 decimal places)
p=p= ____
(report accurate to 4 decimal places)

Conclusion:

• A. The sample data suggest the average starting hourly wages are not the same.
• B. There is not sufficient data to conclude the starting wages are different for the different groups.

A study explored the effect of ethanol on sleep time. Fifteen rats were randomized to one of three treatments. Treatment 1 was water (control). Treatment 2 was 1g of ethanol per kg of body weight, and Treatment 3 was 2g/kg. The amount of REM sleep (in minutes) in a 24hr period was recorded: Treatment 1: 63, 54, 69, 50, 72 Treatment 2: 45, 60, 40, 56 Treatment 3: 31, 40, 45, 25, 23, 28 Are these data strong evidence REM sleep time differs across the three treatment populations?

(a) Graph the data. Why did you choose the graph that you did and what does it tell you?

(b) Create an ANOVA table for the data using your calculator (to prepare for exams). Show your work. You may use R to check your answers.

(c) Evaluate the ANOVA assumptions graphically. Was ANOVA appropriate here?

(d) Based on the ANOVA table, make a conclusion in the context of the problem.

(e) Use R to create 95% CIs for all pairwise comparisons of means using the Tukey-Kramer method. Summarize your results using letter codes. What do you conclude?

Please complete and show all work and R codes for part E only. Thank you!

Suppose your favorite coffee machine offers 7 ounce cups of coffee. The actual amount of coffee put in the cup by the machine varies according to a normal​ distribution, with mean equal to 8 ounces and standard deviation equal to .67 ounces

a. What percentage of cups will be filled with more than 7.7 ounces?

b. What percentage of cups will have in between 7 and 8 ounces of​ coffee?

USING R ---locate the pre-loaded MASS package, then load the data frame cats within that packag. This provides data on sex, body weight (in kgs), and heart weight (in grams) for 144 household cats. Load the MASS package with a call to library("MASS"), and access the object directly by entering cats at the console prompt.

1. Fit a least-squares multiple linear regression model using heart weight as the response variable and the other two variables as predictors, and view a model summary. Write down the equation for a least-squares multiple linear regression fitted model and interpret the estimated regression coefficients for body weight and sex (of the cats in the above package). Are both statistically significant? What does this say about the relationship between the response and predictors?

Consider the game in normal form given in the followingtable. Player 1 is the “row” player with strategiesA,BandCandplayer 2 is the “column” player with strategiesL,CandR. The gameis given in the following table:

(a) Find whether there is a mixed strategy Nash equilibrium (M.S.N.E) where player 1 mixes between A and C and player 2 mixes between L,C and R with positive probability.

(b) Find whether there exists a mixed strategy Nash equilibrium where each player mixes between all her strategies with positive probability.

The F(df1,df2) distribution is negatively skewed. Group of answer choices True False

Taking temperature as an independent variable and pairs of gloves produced as a dependent variable, compute the least square’s regression line.

1). Which of the following statements about the hypothesis testing process is incorrect?

A. The researcher decides on the level of significance after analyzing the output from the statistical test.

B. A decision rule is a statement of the specific conditions when we decide to either reject or not reject the null hypothesis.

C. A test statistic is a value computed from sample information that is used to test the null hypothesis.

D. Holding the significance level and degrees of freedom constant, it is easier to reject the null hypothesis for a mean when conducting a one-tailed test rather than a two-tailed test.

2). Which of the following statements about Type I and Type II is incorrect?

A. When sample size increases, both α and β may decrease.

B. Type I error can only occur when you reject H₀.

C. Type II error can only occur if you fail to reject H₀.

D. The level of significance is the probability of Type II error.

Consider the following hypothesis test.

H₀: μ ≥ 40

H_a: μ < 40

A sample of 36 is used. Identify the p-value and state your conclusion for each of the following sample results. Use

α = 0.01.

(a)

x = 39 and s = 5.3

*Find the value of the test statistic. (Round your answer to three decimal places.)

* Find the p-value. (Round your answer to four decimal places.)

p-value =

*State your conclusion:

Reject H₀. There is sufficient evidence to conclude that μ < 40.

Do not reject H₀. There is sufficient evidence to conclude that μ < 40.    

Do not reject H₀. There is insufficient evidence to conclude that μ < 40.

Reject H₀. There is insufficient evidence to conclude that μ < 40.

Are births really evenly distributed across the days of a week? Here are data on 700 births in a hospital:

<Step 1>

Null hypothesis: the births are evenly distributed across the days of the week

Research hypothesis: the births are not equally probable on all days of the week?

<Step 2> Choose α = 5%

<Step 3> Test statistic used: χ2 =

Decision : Reject Ho if χ2 is too big.

<Step 4> Calculations and Conclusion

1˚ Arrange the data in the form of a frequency distribution (See the table above).

2˚ Obtain the expected frequency for each day.

3˚ Setup a summary table to calculate the Chi-square value.

4˚ Find the degree of freedom.

5˚ Compare the calculated Chi-square value with the appropriate value from the χ2 Table.

The calculated χ2 value is

1. 7.15

2. 8.27    

3. 9.26   

4. 10.10   

5. 11.92

6. 12.76

7. 13.68

8. 16.42

9. 19.12   

10. 23.86.

Q11: (This continues Q10: 2 marks) Find the p-value of the test.

1. Less than 0.25%

2. Between 0.25% and 0.5%

3. Between 0.5% and 1%

4. Between 1% and 2.5%

5. Between 2.5% and 5%

6. Between 5% and 10%

7. Between 10% and 15%

8. Between 15% and 20%

9. Between 15% and 20%

10. Bigger than 20%.