Hypothesis Testing and Confidence Interval Problems

1. Two teachers give the same standardized statistics test. The results are:

Teacher 1 Mean=75 N=20
Teacher 2 Mean=80 N=15

a. What would be a 95% confidence interval for each of the classes?

b. The true mean for the test is 70, with a standard deviation of 5. Can you draw any conclusions about either of the classes from this information?

c. Is there a significant difference for either class?

2. The probability of getting an item on a test correct, is 0.2. Assume that all items are independent.
a. What is the mean for the distribution of probabilities?

b. On a test with 25 items
i. What is the standard deviation?

ii. What is the probability of having exactly 5 correct answers?

iii. What is the probability of having at least 3 correct answers?

iv. How likely is it that 4 answers would be incorrect?

c. Use the normal approximation for find the probability that if there were 100 items, you would get more than 60 correct.

d. What is the probability that you would get two correct and then two wrong?

3. The pass rate for a test is 60%. A teacher gives the test to 40 students; 75% of them pass.
a. What test would you use to test how well the students did?

b. What is the null hypothesis?

c. Is there a significant difference between the class and the overall group?

4. Greek letters are used in statistics
a. To make it more difficult for students
b. Because the Greeks invented statistics
c. To designate characteristics of populations
d. To designate sample parameters

5. A company institutes new policies to increase job satisfaction. They give a sample of employees a questionnaire to measure job satisfaction, before and after the policies are put into place.
a. 10 employees completed the questionnaire. The average change in their level of satisfaction was 1 out of 100 points, with a standard deviation of 20. What is the p-value?

b. The company then had 1000 employees complete the questionnaire and found that the results were statistically significant. Does this differ from the conclusion in a. Why or why not? As an employee do you think these changes are worthwhile?

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...t sample t test.

Decision rule: Reject the null hypothesis, if the P value is less than the significance level.
Details
Z Test for Differences in Two Means

Data
Hypothesized Difference 0
Level of Significance 0.05
Population 1 Sample
Sample Size 20
Sample Mean 75
Population Standard Deviation 5
Population 2 Sample
Sample Size 15
Sample Mean 80
Population Standard Deviation 5

Intermediate Calculations
Difference in Sample Means -5
Standard Error of the Difference in Means 1.707825
Z-Test Statistic -2.9277

Two-Tailed Test
Lower Critical Value -1.95996
Upper Critical Value 1.959964
p-Value 0.003415
Reject the null hypothesis

Conclusion: the sample provides enough evidence to support the claim that there is significant difference in the mean scores.

2. The probability of getting an item on a test correct, is 0.2. Assume that all items are independent.
a. What is the mean for the distribution of probabilities?
Mean p = 0.20
b. On a test with 25 items
i. What is the standard deviation?
Standard deviation = = =0.08
ii. What is the probability of having exactly 5 correct answers?
Let X be the number of correct answers. X is binomial with probability p =0.20 and n =25. The probability density function of binomial probability is given by
P(X=x)=

The probability for different values of x are given below
x P(X=x)
0 0.003778
1 0.023612
2 0.070835
3 0.135768
4 0.186681
5 0.196015

We need P(X=5) = 0.196015
iii. What is the probability of having at least 3 correct answers?
P(X≥3) = 1-P(X<3) = ...