# Descriptive Statistics, Probability & Hypothesis Testing

When performing hypothesis tests it is necessary that you define the hypotheses, sketch the rejection region, and show computations for the test statistic.

1. The normal annual precipitation (in inches) is given below for 5 different American cities.

12.2 9.9 10.3 13.3 16.9

Find the mean, median, and standard deviation for annual precipitation.

2. The probability that a person is immune to a certain disease is 0.40.

a. What is the probability that 4 people will have the disease in a sample of 12 people?

b. Find the mean number of people who have immunity in a sample of size 12.

c. Find the standard deviation for the same sample.

3. If the capacities of the cranial cavities of a certain population are approximately normally distributed with a mean of 1400 cc and a standard deviation of 125, find the probability that a randomly selected person from the given population will have a cranial cavity capacity greater than 1450 cc.

4. IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. In a random sample of 50, approximately how many people will have IQ scores between 95 and 105?

5. Human body temperatures are normally distributed with a mean of 98.6?F and a standard deviation of 0.72?F. Find the temperature that separates the top 7% from the bottom 93%.

6. In a sample of 49 adolescents who served as the subjects in an immunologic study, one variable of interest was the diameter of skin test reaction to an antigen. The sample mean and sample standard deviation were 21 and 11 mm erythema, respectively. Can it be concluded from these data that the population mean is less than 30? Let ? = 0.05. Assume that the population standard deviation is ? = 12.2.

7. A test of abstract reasoning is given to a random sample of students before and after they completed a formal logic course. The results are given below. Construct a 95% confidence interval for the mean difference between the before and after scores. Is there evidence to suggest the logic course improves abstract reasoning? You may assume that the differences for the dependent samples are normally distributed.

Before 74 83 75 88 84 63 93 84 91 77

After 73 77 70 77 74 67 95 83 84 75

8. To examine the effect of pets in stressful situations, researchers recruited 45 women who said they were dog lovers. Fifteen of the subjects were randomly assigned to each of three groups to do a stressful task alone, with a good friend present, or with their dog present. (The stressful task was to count backwards by 13s or 17s.) The subject's mean heart rate during the task is one measure of the effect of stress. The Minitab ANOVA output for the data is given below.

Minitab -- One-way Analysis of Variance

Analysis of Variance for rate

Source DF SS MS F P

Group 2 2387.7 1193.8 14.08 0.000

Error 42 3561.3 84.8

Total 44 5949.0

Individual 95% CIs For Mean

Based on Pooled StDev

Level N Mean StDev -----+---------+---------+---------+-

control 15 82.524 9.242 (-----*-----)

friend 15 91.325 8.341 (-----*-----)

pet 15 73.483 9.970 (-----*-----)

-----+---------+---------+---------+-

Pooled StDev = 9.208 72.0 80.0 88.0 96.0

a. Do the mean heart rates for the groups appear to show that the presence of a pet or a friend reduces heart rate during a stressful task?

b. What are the values of the ANOVA F-statistic and its p-value? What hypotheses does F test? Briefly describe the conclusions you draw from these data. Did you find anything surprising?

9. Are the smoking habits of teenagers related to their parents' smoking? Below are the results of a survey of teenagers in eight Arizona high schools. Expected values are in parentheses. Perform a hypothesis test to determine if the smoking habits of teenagers are independent of the smoking habits of their parents. Test at the 0.05 level of significance.

10. Archaeopteryx is an extinct beast having feathers like a bird but teeth and a long bony tail like a reptile. Five fossil specimens have preserved both the femur and humerus bones. The measurements (in centimeters) for each bone are given below.

Femur 38 56 59 64 74

Humerus 41 63 70 72 84

Scientists are interested in determining if there is a strong relationship between the lengths of the two bones so that they might be persuaded that the five specimens belong to the same species. The Minitab output for regression is provided below.

Minitab -- Regression

The regression equation is humerus = - 3.66 + 1.20*femur

Predictor Coef StDev T P

Constant -3.660 4.459 -0.82 0.472

femur 1.19690 0.07509 15.94 0.001

S = 1.982 R-Sq = 98.8% R-Sq(adj) = 98.4%

Analysis of Variance

Source DF SS MS F P

Regression 1 998.21 998.21 254.10 0.001

Residual Error 3 11.79 3.93

Total 4 1010.00

a. Make a scatter plot of the data with femur length as the predictor variable.

b. What is the slope of the regression equation?

c. What is the correlation coefficient, r?

d. Predict the length of a humerus for a fossil specimen with a femur that is 45 cm long.

e. Does the data suggest that the length of a femur and the length of a humerus are positively correlated? Test at the 0.01 level of significance. Be sure to explain your reasoning.

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#### Solution Summary

The solution provides step by step method for the calculation of descriptive statistics, binomial and normal probabilities and testing of hypothesis. Formula for the calculation and Interpretations of the results are also included. Interactive excel sheet is included. The user can edit the inputs and obtain the complete results for a new set of data.

Statistics: Frequency distribution, median, mode, mean, sample, correlation coefficient

See attached files.

1. Draw a Frequency Distribution for the following data:

17, 1, 3, 4, 4, 16, 2, 2, 2, 7, 8, 7, 1, 2, 2, 5, 4, 3, 12, 9, 6, 3, 6, 10, 5, 2, 3, 11, 14, 2, & 2.

2. Describe briefly the shape of the Frequency Distribution in problem 1.

3. What is the median of the data in 1 above?

4. What is the mode in 1 above?

5. What is the mean of the data in 1 above?

6. The data in 1 above represent a sample drawn from a larger population. What is your best estimate of the population standard deviation?

7. In a certain city 10% of the residents are over 60 years old. What is the probability of finding no one over 60 in a sample of 10 residents?

8. Establish a 95% CI for m on the basis of the following sample statistics:

n = 100

-x = 10

s = 5

9. You are working with samples of the size 15 from the populations that are appx., normal distributed. What tables would you use to establish CI?

10. You suspect that two brands of soft drink differ in acidity. You measure the acidity of 10 randomly selected bottles of each brand with the following results:

Brand A Brand B

n = 10 n = 10

x = 6.50 x = 6.30

s = 0.05 s = 0.04

Outline an appropriate statistical test. What assumptions must you make, if any? Is the difference between the two brands significant at the 1% level?

11. The power of this test against the alternative µ = 5 is 0.80. Explain this statement:

The following data represent growth rates of plates before and after a chemical treatment. Outline an appropriate statistical test of the theory that the chemical treatment reduces growth rate. What assumptions must you make, if any? Are the results significant at the 5% level?

12. Complete the correlation coefficient between the before and after scores in problem 12.

13. Use a regression equation with the data of problem 12 to predict the after growth rate of a plant with a before growth rate of 3.

14. On the basis of sample data you wish to determine whether the variances of two populations differ. What statistical test will you use? Write the formula.

15. You wish to test the theory that men have higher self esteem than women. On the basis of a questionnaire you classify a group of men and woman into categories of high, medium, and low self-esteem. The results are as follows:

High Medium Low

Men 10 30 10

Women 25 15 10

17. Complete the analysis of Variance below. Are there significant differences between groups? (a = 0.01)

Sum of Squares df Variance estimate F

Total 580 29

Between Groups 175 2

Within groups 405 27

18. What assumptions are required for a two-way analysis of variance?

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