Question

In: Statistics and Probability

1. It is known that 15.2% of the population is left-handed. Select a sample of 60...

1. It is known that 15.2% of the population is left-handed. Select a sample of 60 individuals? a) Describe the sampling distribution of the sample proportion. Be sure to calculate the mean (expected value) and the standard deviation of the sample proportion. b) What is the probability that at least 20% of individuals in this sample will be left-handed? c) What is the probability that at most 35% of individuals in this sample will be left-handed?

Solutions

Expert Solution

Solution :

a) If p is a population proportion and a sample of size n is taken from this population and if np ≥ 5, nq ≥ 5, then sampling distribution of sample proportions (p̂) of all the samples of size n will be approximately normal with mean p and standard deviation .

i.e.

(Where, q = 1 - p)

Population proportion of left-handed (p) = 15.2% = 0.152

Sample size (n) = 60

q = 1 - 0.152 = 0.848

np = 60 × 0.152 = 9.12 which is greater than 5.

no = 60 × 0.848 = 50.88 which is greater than 5.

Hence, sampling distribution of sample proportion will be approximately normal with mean p and standard deviation .

The mean of the sample proportions is given as follows:

The mean of the sample proportions is 0.152.

The standard deviation of the sample proportions is given as follows :

The standard deviation of the sample proportions is 0.04635.

Hence, sampling distribution of sample proportion would be approximately normal with mean 0.152 and standard deviation 0.04635.

b) We have to obtain Pr(p̂ ≥ 0.20).

We have, p = 0.152, q = 0.848 and n = 60

We know that if ~ N(p, pq/n) then,

Using "pnorm" function of R we get, Pr(Z ≥ 1.0356) = 0.1502

Hence, the probability that at least 20% of individuals in this sample will be left-handed is 0.1502.

c) We have to obtain Pr(p̂ ≤ 0.35).

We have, p = 0.152, q = 0.848 and n = 60

We know that if ~ N(p, pq/n) then,

Using "pnorm" function of R we get, Pr(Z ≤ 4.2719) = 1.0000

Hence, the probability that at most 35% of individuals in this sample will be left-handed is 1.0000.

Please rate the answer. Thank you.


Related Solutions

1. Given a population in where the probability of someone being left handed is 0.3, people...
1. Given a population in where the probability of someone being left handed is 0.3, people is taken then... Calculate the probability the proportion of left-h
1. Assume that 7% of people are left-handed. If we select 5 people at random, find...
1. Assume that 7% of people are left-handed. If we select 5 people at random, find the probability of each outcome described below, rounded to four decimal places: a. There are some lefties ( ≥ 1) among the 5 people. b. There are exactly 3 lefties in the group. c. There are at least 4 lefties in the group. d. There are no more than 2 lefties in the group. e. How many lefties do you expect? f. With what...
Approximately 13% of the population are left-handed. If three people are randomly selected, what is the...
Approximately 13% of the population are left-handed. If three people are randomly selected, what is the probability that all are left- handed?   (round to 4 decimal places) If three people are randomly selected, what is the probability that at least 1 person is not left handed?   (round to 4 decimal places) Notice that these two events are complementary (the probabilities do add up to 1).
Suppose the proportion of left-handed individuals in a population is θ. Based on a simple random...
Suppose the proportion of left-handed individuals in a population is θ. Based on a simple random sample of 20, you observe four left-handed individuals. Using R a) Assuming the sample size is small relative to the population size, plot the loglikelihood function and determine the Maximum Likelihood Estimate. b) If instead the population size is only 50, then plot the log-likelihood function and determine the MLE. (Hint: Remember that the number of left-handed individuals follows a hypergeometric distribution. This forces...
Assume that 73% of people are left-handed. If we select 5 people at random, find the...
Assume that 73% of people are left-handed. If we select 5 people at random, find the probability of each outcome described below, rounded to four decimal places: a. There are some lefties (≥ 1) among the 5 people. b. There are exactly 3 lefties in the group. c. There are at least 4 lefties in the group. d. There are no more than 2 lefties in the group. e. How many lefties do you expect? f. With what standard deviation?
The probability that a person is left-handed is 17%. You randomly select three people. What is...
The probability that a person is left-handed is 17%. You randomly select three people. What is the probability that: a) all three are left handed b) none of them are left handed c) at least one of them is left-handed
Assume that 37% of people are left-handed. If we select 5 people at random, find the...
Assume that 37% of people are left-handed. If we select 5 people at random, find the probability of each outcome described below, rounded to four decimal places: a. There are some lefties ( ≥ 1) among the 5 people. b. There are exactly 3 lefties in the group. c. There are at least 4 lefties in the group. d. There are no more than 2 lefties in the group. e. How many lefties do you expect? f. With what standard...
Assume that 51% of people are left-handed. If we select 5 people at random, find the...
Assume that 51% of people are left-handed. If we select 5 people at random, find the probability of each outcome described below, rounded to four decimal places: a. There are some lefties (≥ 1) among the 5 people. b. There are exactly 3 lefties in the group. c. There are at least 4 lefties in the group. d. There are no more than 2 lefties in the group. e. How many lefties do you expect? f. With what standard deviation?
Assume that 57% of people are left-handed. If we select 5 people at random, find the...
Assume that 57% of people are left-handed. If we select 5 people at random, find the probability of each outcome described below, rounded to four decimal places: a. There are some lefties (≥ 1) among the 5 people. b. There are exactly 3 lefties in the group. c. There are at least 4 lefties in the group. d. There are no more than 2 lefties in the group. e. How many lefties do you expect? f. With what standard deviation?
Assume that 41% of people are left-handed. If we select 5 people at random, find the...
Assume that 41% of people are left-handed. If we select 5 people at random, find the probability of each outcome described below, rounded to four decimal places: a. There are some lefties (≥ 1) among the 5 people. b. There are exactly 3 lefties in the group. c. There are at least 4 lefties in the group. d. There are no more than 2 lefties in the group. e. How many lefties do you expect? f. With what standard deviation?
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT