Lab Activity 8 Hypothesis Testing Data Set: Spring17Data.MTW Number of Questions

Question

Lab Activity 8

Hypothesis Testing

Data Set: Spring17Data.MTW

Number of Questions: 3

Due: March 19, 2017

1 Hypothesis Testing for a Mean Using Summarized Data – Errors and Power

A test prep company has developed a new intervention to improve SAT-Verbal scores. They will conduct a hypothesis test to determine if the average SAT-Verbal score of a student who has completed their program is greater than the known national average of 490. The program is very expensive at $12,000 per student. Use this scenario to answer the following questions.

Write the null and alternative hypotheses using the appropriate symbols.

Ho:

Ha:

b. In terms of this scenario, describe what a Type I error would look like. (Be sure to apply this scenario!)

c. In terms of this scenario, describe what a Type II error would look like. (Be sure to apply this scenario!)

d. The researchers used a sample size of n = 5,000 and found a sample mean of 495. Using the hypotheses from part a, and the known population standard deviation of 98, what is the test statistic? You may do hand calculations or use software.   Remember to show all work and/or output for credit!

e. What is the p-value associated with the test statistic you computed in part d? You can use the t-table or software. Remember to show all work and/or output for credit!

f. Are the results of this study statistically significant? 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 $12,000 to send your child through this program? Explain why or why not.

h. Run the analyses from part d again, this time using a sample size of n = 30. Use the five step hypothesis testing procedure outlined below. Remember to show all work and/or output for credit!

Step 1: Write the null and alternative hypotheses:

Null:                   

Alternative:

Step 2: Calculate the test statistic:

Step 3: Find the p-value:

Step 4: Decide between the null and alternative hypothesis:

Step 5: State a real world conclusion:

i. How did decreasing the sample size from 5,000 to 30 change the statistical power of this test?

2 Hypothesis Testing for a Mean Using Raw Data

Answer the following questions using the dataset Spring17Data.MTW. Consider this data to be a sample that is representative of the population of all Penn State World Campus STAT 200 students.

At the 0.05 alpha level, is there evidence that the mean weight of all Penn State World Campus STAT 200 students is greater than 170 pounds? Use the five step hypothesis test procedure outlined below. You can use software for the analysis. Be sure to include all output! Remember to show all work and/or output for credit!

Step 1: Write the null and alternative hypotheses:

Null:            

Alternative:

Step 2: Calculate the test statistic:

Step 3: Find the p-value:

Step 4: Decide between the null and alternative hypothesis:

Step 5: State a real world conclusion:

In part a, is it possible that a Type I or Type II error was committed? Explain.

3 Hypothesis Testing for a Proportion Using Raw Data

Answer the following questions using the dataset Spring17Data.MTW. Consider this data to be a sample that is representative of the population of all Penn State World Campus STAT 200 students.

At the 0.05 alpha level, is there evidence that the proportion of students eating breakfast in the population of all Penn State World Campus STAT 200 students is different from 60%? Use the five step hypothesis test procedure outlined below. If appropriate, apply the normal approximation method. Remember to show all work and/or output for credit!

Step 1: Write the null and alternative hypotheses:

Null:                

Alternative:

Step 2: Calculate the test statistic:

Step 3: Find the p-value:

Step 4: Decide between the null and alternative hypothesis:

Step 5: State a real world conclusion:

Construct a 95% confidence interval for the proportion of the population that eats breakfast. Use the normal approximation method if appropriate. Remember to show work and/or output for credit!

Explain how the results from both parts a and b could be used to answer the same research question.

Explanation / Answer

a. Null Hypothesis: Average SAT-Verbal score of a student who has completed the program is equal to the national average of 490.

Alternative Hypothesis: Average SAT-Verbal score of a student who has completed the program is greater than the national average of 490.

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-Verbal score of a student who has completed the program is greater than the national average of 490 but in reality, average SAT-Verbal score of a student who has completed the program is less or equal to the national average of 490.

c. b. 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-Verbal score of a student who has completed the program is less or equal to the national average of 490 but in reality, average SAT-Verbal score of a student who has completed the program greater than the national average of 490.

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

observed value = 495

hypothesized value = 490

standard error = standard deviation/ sqrt(number of samples) = 98/sqrt(5000) = 1.386

test statistic = (495 - 490)/1.386 = 3.607

e. degree of freedom = number of samples - 1 = 5000 - 1 = 4999

p-value of test statistic, 3.607 with degree of freedom, 4999 = 0.00015

f. Assuming 95 % confidence interval (0.05 significance level), 0.00015 < 0.05, we reject the null hypothesis and accept the alternative hypothesis that Average SAT-Verbal score of a student who has completed the program is greater than the national average of 490. 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, 4999 is 1.64 (Calculated from t-test table)

Let the average SAT-Verbal score be x. then

(x - 490)/1.386 = 1.64

or, x - 500 = 2.273

or, x = 502.273

So, we are 95% confident that the avearge SAT-Verbal score would be 502.27 when the student has completed the program.

The results of this study does not seems to be practically significant as spending $12,000 would only increase the average score by 2.27 than the national average.

h. Sample Size, n = 30

standard error = standard deviation/ sqrt(number of samples) = 98/sqrt(30) = 17.892

test statistic = (495 - 490)/17.892 = 0.2794

degree of freedom = number of samples - 1 = 30 - 1 = 29

p-value of test statistic, 0.2794 with degree of freedom, 29 = 0.391

Assuming 95 % confidence interval (0.05 significance level), 0.391 > 0.05, we fail to reject the null hypothesis and accept the alternative hypothesis that Average SAT-Verbal score of a student who has completed the program is greater than the national average of 490.

i.

When the sample size is 5000, there is probability of 0.00015 that the average SAT-Verbal score is not greater than 490.

When the sample size is 30, there is probability of 0.391 that the average SAT-Verbal score is not greater than 490. 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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