# Stats Math 3

Suppose a simple random sample of size n=150 is obtained from a population whose size is N=20,000 and whose population proportion with a specified characteristic is p=0.8. Please answer questions (1) through (5) below.

(1) Describe the sampling distribution of p^ (choose the correct phrase that best describes the shape of the sampling below)

a) Not normal because n <, 0.05N and np (1-p) <10.

b) Approximately normal because n <, 0.05N and np (1-p) >10.

c) Approximately normal because n <, 0.05N and np (1-p) <10.

d) Not normal because n <, 0.05N and np (1-p) >10.

2) Determine the mean of the sampling distribution of p^.

p^ = ______ (round to one decimal place as needed)

3) Determine the standard deviation of the sampling distribution of p^.

p^ = ______ (round six decimal places as needed)

4) What is the probability of obtaining x=123 or more individuals with the characteristic? that is , what P(p^ > 0.82)?

P( p^ > 0.82) = _____ (Round four decimals places)

5) What is the probability of obtaining x = 111 of fewer individuals with the characteristic ? that is what is P(p^ < 0.74)?

P(p^ < 0.74) = _______ (Round to four decimal places as needed)

1. 👍 0
2. 👎 0
3. 👁 2,260
1. (1)
The sample distribution taken from a binomial distribution is approximately normal because sample/population size < 0.05 ("small" sample). Also a binomial distribution approximates a normal distribution when np>K and np(1-p)>K, where K ranges between 5 & 10, depending on preference.

(2)
The expected value of p̂, or E[p̂] is the mean of the population. The mean of a binomial population is np.

(4)
P(X≥123)=ΣP(Xi),i=123...n
where P(Xi)=C(n,i)pi(1-p)(n-i)
and C(n,i)=n!/(i!(n-i)!) is combination of i objects from n.
It's a relative long process to do the summation. You can use binomial tables, or you can approximate the value using the normal approximation to binomial distribution, as hinted in part (1).

(5) see (4)

1. 👍 0
2. 👎 3

## Similar Questions

1. ### statistics

Suppose a random sample of size 50 is selected from a population with σ = 10. Find the value of the standard error of the mean in each of the following cases (use the finite population correction factor if appropriate). a. The

2. ### Economics (39)

Suppose a random sample of size 40 is selected from a population with = 9. Find the value of the standard error of the mean in each of the following cases (use the finite population correction factor if appropriate). a. The

3. ### statistics

A simple random sample of 60 items resulted in a sample mean of 80. The population standard deviation is 15. a) Compute the 95 % confidence interval for the population mean. b) Assume that the same sample mean was obtained from a

4. ### stats

Three firms carry inventories that differ in size. Firm A's inventory contains 2000 items, firm B's inventory contains 5000 items, and firm C's inventory contains 10,000 items. The population standard deviation for the cost of the

1. ### Statistics 1 plaese give answers to below question

Suppose a simple random sample of size n=150 is obtained from a population whose size is N=20,000 and whose population proportion with a specified characteristic is p=0.8. Please answer questions (1) through (5) below. (1)

2. ### Statistics (42)

A simple random sample of 60 items resulted in a sample mean of 96. The population standard deviation is 16. a. Compute the 95% confidence interval for the population mean (to 1 decimal). ( , ) b. Assume that the same sample mean

3. ### statistics

a simple random sample of size n=5 is obtained from the population of drivers living in New York City, and the breaking reaction time of each driver is measured. The results are to be used for constructing a 95% confidence

A random sample of size 15 taken from a normally distributed population reveled a sample mean of 75 and a sample variance of 25. The upper limit of a 95% confidence interval for the population mean would equal?

1. ### Math (Statistic)

Considered the sampling distribution of a sample mean obtained by random sampling from an infinite population. This population has a distribution that is highly skewed toward the larger values. a) How is the mean of the sampling

2. ### Statistics

A new manager, hired at a large warehouse, was told to reduce the 26% employee sick leave. The manager introduced a new incentive program for the company's employees with perfect attendance. The manager decides to test the new

3. ### Statistics

You are interested in estimating the mean of a population. you plan to take a random sample from the population and use the samples mean as an estimate of population mean. Assuming that the population from which you select your

4. ### Stats

Suppose a random sample of size 58 is selected from a population with σ = 9. Find the value of the standard error of the mean in each of the following cases (use the finite population correction factor if appropriate). A) The