Answer the following questions showing all work. Full credit will not be given t

Question

Answer the following questions showing all work. Full credit will not be given to answers without work shown. If you use Minitab Express include the appropriate output (copy + paste) along with an explanation. Output without explanation will not receive full credit. Use equation editor to communicate correct notation. Round all answers to 3 decimal places. If you have any questions, post them to the course discussion board.

1. A test prep company has developed a new intervention to improve SAT-Math scores. They will conduct a hypothesis test to determine if the average SAT-Math score of a student who has completed their program is greater than the known national average of 500. The program is very expensive at $4,000 per student. Use this scenario to answer the following questions. [40 points]

A. Write out the null and alternative hypotheses using the appropriate symbols. For a review of using the equation editor in Word, see https://onlinecourses.science.psu.edu/statprogram/equations

B. In terms of this scenario, describe what a Type I error would result in. (Be sure to discuss this contextually)

C. In terms of this scenario, describe what a Type II error would result in. (Be sure to discuss this contextually)

D. The researchers used a sample size of n= 800 and found a sample mean of 508. Using your hypotheses from part A and the known population standard deviation of 100, what is the test statistic? You may do hand calculations or use Minitab Express (https://onlinecourses.science.psu.edu/stat200/node/228)

E. What is the p-value associated with the test statistic you computed in part D? Show correct notation here.

F. Are the results of this study statistically significant ( = 0.05) ? Explain why.

G. Are the results of this study practically significant? For example, if you were a parent, would you be convinced that you should spend $4,000 to send your child through this program? Explain why.

H. Rerun the analyses from part D using a sample size of n=80. Use the five-step hypothesis testing procedure. Step 1: Step 2: Step 3: Step 4: Step 5:

I. How did decreasing the sample size from 800 to 80 change the statistical power of this test? Explain in qualitative terms…you are NOT expected to quantify this.

2. Use the dataset SURVEY_DATA_STAT200_S17.MTW. Consider this data to be a sample that is representative of the population of all Penn State World Campus STAT 200 students. [20 points]

A. At the 0.05 alpha level, is there evidence that the mean exercise minutes per week of all Penn State World Campus STAT 200 students is less than 180 minutes ( 3 hrs)? Use the five-step hypothesis testing procedure. Use Minitab Express to automate this test, supply screen output in the necessary step(s) below. Step 1: Step 2: Step 3: Step 4: Step 5: B. In part A, is it possible that a Type I and/or Type II error was committed? Explain why.

3. Use the dataset SURVEY_DATA_STAT200_S17.MTW. Consider this data to be a sample that is representative of the population of all Penn State World Campus STAT 200 students. [25 points]

A. At the 0.01 alpha level, is there evidence that the true proportion of students that are hunters is different than 40%? Use the five-step hypothesis testing procedure. Use the normal approximation method if appropriate. Use Minitab Express to automate this test, supply screen output in the necessary step(s) below. Step 1: Step 2: Step 3: Step 4: Step 5:

B. Construct and interpret a 99% confidence interval for the proportion of the population that consider themselves hunters. Use the normal approximation method if appropriate. Use Minitab Express to automate this test, supply screen output in the necessary step(s) below.

C. Do the results of part (A) and (B) agree with each other? Explain. 4. Use the dataset SURVEY_DATA_STAT200_S17.MTW. Consider this data to be a sample that is representative of the population of all Penn State World Campus STAT 200 students. [15 points] A marketing agency would like to estimate the true population mean price that students believe is reasonable for the total cost of a hoagie and a soda. They would like to do this with a 95% ‘assuredness’. Using what you have learned to this point in the course, perform the appropriate statistical inference. Be sure to introduce your process, execute the correct mathematics (from the survey data), and interpret your findings specifically referring to your quantitative results. Use Minitab Express as necessary.

Explanation / Answer

A. Null Hypothesis: Average SAT-Math score of a student who has completed the program is equal to the national average of 500.

Alternative Hypothesis: Average SAT-Math score of a student who has completed the program is greater than the national average of 500.

B. Type I error - A type I error occurs when one rejects the null hypothesis when it is true. A type I error occurs when we determine that the average SAT-Math score of a student who has completed the program is greater than the national average of 500 but in reality, average SAT-Math score of a student who has completed the program is less or equal to the national average of 500.

C. Type II error - A type II error occurs when one rejects the alternative hypothesis (fails to reject the null hypothesis) when the alternative hypothesis is true. A type II error occurs when we determine that the average SAT-Math score of a student who has completed the program is less or equal to the national average of 500 but in reality, average SAT-Math score of a student who has completed the program is greater than the national average of 500.

D. test statistic = (observed value - hypothesized value)/standard error

observed value = 508

hypothesized value = 500

standard error = standard deviation/ sqrt(number of samples) = 100/sqrt(800) = 3.53

test statistic = (508 - 500)/3.53 = 2.266

E. degree of freedom = number of samples - 1 = 800 - 1 = 799

p-value of test statistic, 2.266 with degree of freedom, 799 = 0.0118 (Calculated from t-test table)

F. Assuming 95 % confidence interval (0.05 significance level), 0.0118 < 0.05, we reject the null hypothesis and accept the alternative hypothesis that Average SAT-Math score of a student who has completed the program is greater than the national average of 500. Yes, the results are statistically significant as the p-values are less than 0.05.

G. At 95%, the t-value for degree of freedom, 799 is 1.64 ((Calculated from t-test table)

Let the average SAT-MAth score be x. then

(x - 500)/3.53 = 1.64

or, x - 500 = 5.79

or, x = 505.79

So, we are 95% confident that the avearge SAT-Math score would be 505.79 when the student has completed the program.

The results of this study does not seems to be practically significant as spending $4000 would only increase the average score by 5.79 than the national average.

H. Sample Size, n = 80

standard error = standard deviation/ sqrt(number of samples) = 100/sqrt(80) = 11.18

test statistic = (508 - 500)/11.18 = 0.7155

degree of freedom = number of samples - 1 = 80 - 1 = 79

p-value of test statistic, 0.7155 with degree of freedom, 79 = 0.2382

Assuming 95 % confidence interval (0.05 significance level), 0.2383 > 0.05, we fail to reject the null hypothesis and reject the alternative hypothesis that Average SAT-Math score of a student who has completed the program is greater than the national average of 500.

I. When the sample size is 800, there is probability of 0.0118 that the average SAT-Math score is not greater than 500.

When the sample size is 80, there is probability of 0.2383 that the average SAT-Math score is not greater than 500. So, the probability of commiting type II error increases when the sample size decreases and consequently, the statistical power of the test decreases when the sample size decreases.

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