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Probability Distribution: Binomial & Normal

Unit 2 Test

1. The United States National Centre for Education Statistics compiles enrolment data on American public schools and reports the information in Digest of Education Statistics. The following table displays a frequency distribution for the enrolment by grade level in public secondary schools for a given year. Frequencies are in thousands.
Grade, x Frequency, f
9 3604
10 3131
11 2749
12 2488
Total 11972

(a). Determine the probability distribution of the random variable x. (Hint: complete the table by finding relative frequencies.)
(b). Construct a probability histogram for the random variable x.
(c). What is the probability that a randomly selected student is in either 9th or 10th grade?
(e). What is the probability that a randomly selected student is in at most 11th grade?
(f). Find the mean and standard deviation of the random variable x.

2. Shaquille O'Neal is a popular basketball player, but is not recognized as a great free throw shooter. His career free throw average is 0.532 (in other words he has made approximately 53.2% of his free throws). Suppose in a particular game that Shaq takes 5 free throws.

a. What type of probability distribution is this? Explain.
b. What are the mean and standard deviation for the number of made free throws?
c. What is the probability that he will miss 4 of the next 5 free throws he takes?

3. Consider two normally distributed random variables, X and Y, such that

µx = 2 and σx = 5

µy = 2 and σy = 1

In other words, both variables have the same mean but different standard deviations. Draw rough sketches of the two normal curves on the same graph. Be sure to label your curves.

4. Determine the area under the standard normal curve corresponding to the following regions. Be sure to draw a picture showing the specified region.
a. the region to the left of z = 1.3
b. the region between z = -1.42 and z = .07
c. the region to the right of z = 2.45
d. the region between z = -1.96 and z = 1.96

5. The average length of stay in a chronic disease hospital for a certain type of patient is 60 days with a standard deviation of 15. Suppose it is reasonable to assume an approximately normal distribution of lengths of stay.

a. What percentage of patients stays less than 50 days?
b. Fill in the blanks.
The probability is 0.9534 that a patient will stay between __________ and __________ days.

6. In the study of fingerprints, an important quantitative characteristic is the total ridge count for the 10 fingers of the individual. Suppose that the total ridge counts of individuals in a certain population are approximately normally distributed with a mean of 140 and a standard deviation of 50.

a. Find the probability that an individual picked at random from this population will have a ridge count of 200 or more.
b. In a population of 10,000 people how many would you expect to have a ridge count of 200 or more?

7. The National Health and Nutrition Examination Survey of 1976-80 found that the mean serum cholesterol level for U.S. males aged 20-74 years was 211. The standard deviation was approximately 90. Consider the sampling distribution of the sample mean based on samples of size 100 drawn from this population of males.

a. What are the mean and standard deviation of the sampling distribution?
b. Is it necessary for the original distribution to be normal? Explain.
c. Suppose a random sample of 100 is taken, what is the probability that the mean serum cholesterol level will be between 198 and 220?

See the attached file.

Attached is a copy of the unit notes and the assignment/test is on page 15-17. Please do not use minitab.


Solution Summary

The solution provides step by step method for the calculation of binomial and normal probabilities. Formula for the calculation and Interpretations of the results are also included.