Statistics and Probability Answers

A local paper claims the proportion of college students who own a car is 38%. In a sample of 275 college students, what is the probability that more than 116 will own a car?

sketch in the normal table and determine the probability of the following z scores

a. p(z<+0.85)

b. p(z > - 1.23)

c. p(-1.50 <z< + 1.50)

d. p(0.00 <z< + 2.27)

Suppose that a Covid-19 testing centre receives two phones calls, on average, per minute concerning test results.

Required:

a) What are the conditions for this experiment to be considered a Poisson experiment? Motivate. (4)

b) What is the expected number of phone calls regarding test results per hour? (2)

c) What is the probability that more zero but less than four phone calls regarding test results are received in any given period of two minutes? (9)

d) What is the probability that more than one phone calls regarding test results are received in any given period of three minutes? (7)

e) What is the probability that no phone calls regarding test results are received in any given period of one minute? (3) 

Random samples of size n=3 are drawn of a finite population consisting of the numbers 5, 6, 7, 8, and 9.

a. How possible samples are there?

b. List all the possible samples and the corresponding mean for each sample.

c. Construct and Sampling distribution of the sample means.

d. Construct the histogram for the sampling distribution of the sample mean.                   Describe the shape of the histogram.

If a population with size 3 has a standard deviation of 2.4, what is thestandard deviation of the sampling distribution of its means? The sampling distribution has a sample size of 2 and all possible samples are drawn without replacements. 

Let (Z₁, Y₁), . . . , (Z_n, Y_n) be generated as follows:

Z_i ∼ Bernoulli(p)

Y_i ∼ { N(0, 1) if Z_i = 0 , N(5, 1) if Z_i = 1

(a) Assume we do not observe the Z_i ’s. Write the pdf f(y) of Y as a mixture of two normal distribution pdf. (Use the notation φ(·) which is the standard normal pdf.)

(b) Write down the likelihood function for p (without Zi ’s).

(c) Write down the complete likelihood function for p (assuming the Zi ’s are observed).

(d) Find the maximum likelihood estimation of p using the likelihood from (c). 

Suppose we have the following data:

30

79

59

65

40

64

52

53

57

39

61

47

50

60

48

50

58

67

Suppose the number of nonoverlapping classes is determined to be 5. 

What is the mean and sample variance for this data?

a biologist estimates that 50% of deer in the region carry a certain type of tick. for a sample of 300 deer selected at random, what is the chance that 155 or fewer deer have this tick?

The average number of pages in a novel is 326 with a standard deviation of 24

pages. If a sample of 50 novels is randomly chosen, what is the probability that

the average number of pages in these books is between 319 and 331?

On a given afternoon, the probability that a certain manager will be in her office is 0.48 and the probability that she will be in her home is 0.27. Assuming that the manager’s office and home are in two different locations, find the probability that she will be neither in her office nor in her home on a given afternoon.