Question

In: Statistics and Probability

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 θ to be of the form i/50 for some integer i between 4 and 34. From a tabulation of the log-likelihood, you can obtain the MLE.)

Solutions

Expert Solution

a) When the sample size is small relative to the population size, the number of left-handed individuals

follows Binomial distribution.

The PMF of X is

The likelihood for observation is

Differentiating with respect to and equating to 0,

b) The probability of observing left-handed individuals is

. Where

Also

The loglikelihood is printed below:

           [,1]
 [1,]  -3.861436
 [2,]  -2.679442
 [3,]  -2.020196
 [4,]  -1.624883
 [5,]  -1.397099
 [6,]  -1.288886
 [7,]  -1.272756
 [8,]  -1.331597
 [9,]  -1.454199
[10,]  -1.633018
[11,]  -1.862941
[12,]  -2.140573
[13,]  -2.463800
[14,]  -2.831525
[15,]  -3.243505
[16,]  -3.700263
[17,]  -4.203057
[18,]  -4.753888
[19,]  -5.355563
[20,]  -6.011806
[21,]  -6.727426
[22,]  -7.508584
[23,]  -8.363181
[24,]  -9.301451
[25,] -10.336884
[26,] -11.487747
[27,] -12.779731
[28,] -14.251018
[29,] -15.963314
[30,] -18.031327
[31,] -20.739377

The loglikelihood is plotted below.

:

The likelihood is maximized at . So the MLE is

R code below:

X <- 4:34
N <- 50
P <- choose(X,4)*choose(N-X,16)/choose(N,20)
L <- log(P)
ML <- matrix(L,nrow=31,ncol=1)
ML
plot(X, L, type="l", col="blue", main="Loglikelihood")


Related Solutions

How would one set up a survey to compare the proportion of left-handed men to left-handed...
How would one set up a survey to compare the proportion of left-handed men to left-handed women? A layout example would be appreciated
Researchers wanted to compare the proportion of females who are left-handed to the proportion of males...
Researchers wanted to compare the proportion of females who are left-handed to the proportion of males who are left-handed. The researchers randomly selected 815 females and 575 males. They observed that 66 of the females and 60 of the males were left-handed. Test the claim that the proportion of females who are left-handed is lower than the proportion of males who are left-handed. Test the claim at the 5% significance level. Credit will be given for a complete analysis that...
Researchers wanted to compare the proportion of females who are left-handed to the proportion of males...
Researchers wanted to compare the proportion of females who are left-handed to the proportion of males who are left-handed. The researchers randomly selected 815 females and 575 males. They observed that 66 of the females and 60 of the males were left-handed. Test the claim that the proportion of females who are left-handed is lower than the proportion of males who are left-handed. Test the claim at the 5% significance level.
1. Assume that 13​% of people are​ left-handed. Suppose 8 people are selected at random. Answer...
1. Assume that 13​% of people are​ left-handed. Suppose 8 people are selected at random. Answer each question about​ right-handers below. ​a) Find the mean and standard deviation of the number of​ right-handers in the group. ​b) What's the probability that​ they're not all​ right-handed? ​c) What's the probability that there are no more than 5 ​righties? d) What's the probability that the majority is​ right-handed? 2. A certain tennis player makes a successful first serve 64​% of the time....
You want to know if right-handed individuals have a lower grade point average than left handed...
You want to know if right-handed individuals have a lower grade point average than left handed individuals. Question 1 options: Multiple Regression chi squared t-test correlation ANOVA (F test) Question 2 (1 point) Saved You want to see if a new ad campaign toward underage drinking is working. You break people up into two groups (over 21 and under 21) and measure the number of alcoholic beverages they have had in the past 2 weeks. Question 2 options: chi squared...
Given: "The proportion of left-handed people is equal to 10%" a) Provide a Conclusion that would...
Given: "The proportion of left-handed people is equal to 10%" a) Provide a Conclusion that would be a Type I error b) Provide a Conclusion that would be a Type II error
In a simple random sample of size 50, taken from a population, 23 of the individuals...
In a simple random sample of size 50, taken from a population, 23 of the individuals met a specified criteria. a) What is the margin of error for a 90% confidence interval for p, the population proportion? Round your response to at least 3 decimal places.     b) What is the margin of error for a 95% confidence interval for p? Round your response to at least 3 decimal places.    
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?
A simple random sample of size n=69 is obtained from a population that is skewed left...
A simple random sample of size n=69 is obtained from a population that is skewed left with u=48 and o=6. Does the population need to be normally distributed for the sampling distribution of x to be approximately normally​ distributed? Why? What is the sampling distribution of x​?
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).
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT