Answer to Question #168010 in Statistics and Probability for Paul

Question

1). There are three insurance companies in the Business District in Accra. It is known from previous records that each year, 2 percent, 5 percent and 3 percent of those insured by Vanguard Insurance, Enterprise Insurance and Star insurance respectively submit claims. 20 percent of the people in the community have an insurance policy with Vanguard Insurance Company, 30 percent with Enterprise Insurance Company and 50 percent with Star Insurance Company.

(i) What is the probability that a randomly selected insurance policyholder in the community did a claim in a certain year?

(ii) If a randomly selected policy holder in the community is found to have made a claim in a certain year, what is the probability that, that person is insured with Vanguard Insurance?

(iii) If a randomly selected policy holder in the community is found to have made a claim in a certain year, what is the probability that, that person is insured with Enterprise Insurance?

2 (a). Researchers are concerned about the impact of students working while they are enrolled in classes, and they’d like to know if students work too much and therefore are spending less time on their classes than they should be. First, the researchers need to find out, on average, how many hours a week students are working.

They know from previous studies that the standard deviation of this variable is about 5 hours.

(i) A survey of 200 students provides a sample mean of 7.10 hours worked. 95% confidence interval based on this sample is 6.41 to 7.79. Do you agreed with the researcher that the 95% confident interval of the unknown population mean of the students worked hours lies between 6.41 and 7.79? Indicate whether the researchers are right or wrong, giving reasons for your answer.

(ii) Based on your understanding of confident interval estimation; explain the reason why the researcher did not use 100% confidence level, but instead, he used 95% confidence level.

(iii) The researchers are not satisfied with their confidence interval and want to do another study to find a shorter confidence interval. What should they change to ensure they find a shorter confidence interval?

3) The average lifespan of a Ghanaian is 57years. The average lifespan of males in Ghana is known to be 55 years while the average lifespan of females in Ghana is known to be 60 years. The average lifespan of people from the Northern region of Ghana is known to be 58 years while the average lifespan of people in the East region of Ghana is known to be 54 years. Use the above information to answer the following questions:

(I). With your reasons, discuss all possible population parameters from the above information.

(ii). With your reasons, discuss all possible sample statistics from the above information.

(iii) Compare and contrast the average lifespan of males and CR(10marks) 4 females. Which group average is more likely to represent the corresponding population parameter? Give reasons for your answer.

4) In each of the following research situations below, determine whether the data should be analyzed with a parametric test, a nonparametric test, or both. Give reasons for your answer in each situation.

(i). You want to compare males and females scores obtained from an aptitude test where scores on the aptitude test range between 0 and 100.

(ii). You want to find out whether or not basketball players are taller than the general population.

(iii). You want to find out whether or not a difference exists between males and females in their like or dislike for statistics.

(iv). You want to find out among a sample of eligible voters in the Ghanaian population, whether the political party they voted for in the 2016 general elections was the same as the political party they voted for in the 2012 general elections.

(v).You want to compare two groups of university students on their levels of intelligence, where intelligence scores could range between 30 and 150

Answer 1

$1)\ (i)\\ H_1\text{ --- selected Vanguard Insurance holder}.\\ H_2\text{ --- selected Enterprise Insurance holder}.\\ H_3\text{ --- selected Star Insurance holder}.\\ A\text{ --- selected a policy holder who made a claim in a certain year}.\\ \text{Using the Law of Total probability:}\\ P(A)=P(H_1)P(A|H_1)+P(H_2)P(A|H_2)+P(H_3)P(A|H_3)=\\ =(0.2)(0.02)+(0.3)(0.05)+(0.5)(0.03)=0.034.\\ (ii)\\ \text{Using Bayes' theorem:}\\ P(H_1|A)=\frac{P(H_1\cdot A)}{P(A)}=\frac{P(H_1)\cdot P(A|H_1)}{P(A)}=\\ =\frac{(0.2)\cdot (0.02)}{0.034}\approx 0.12.\\ (iii)\\ \text{Using Bayes' theorem:}\\ P(H_2|A)=\frac{P(H_2\cdot A)}{P(A)}=\frac{P(H_2)\cdot P(A|H_2)}{P(A)}=\\ =\frac{(0.3)\cdot (0.05)}{0.034}\approx 0.44.\\ 2)\ (i)\\ \sigma=5\\ N=200\\ \overline{x}=7.10\\ \gamma=95\%\\ (\overline{x}-\delta;\overline{x}+\delta)\\ \delta=\frac{t_{\gamma}\sigma}{\sqrt{n}}\\ 2\Phi(t_{\gamma})=\gamma\\ 2\Phi(t_{\gamma})=0.95\\ \Phi(t_{\gamma})=0.475\\ t_{\gamma}\approx 1.96\\ \delta=\frac{(1.96)\cdot 5}{\sqrt{200}}\approx 0.69\\ (7.10-0.69;7.10+0.69)\\ (6.41;7.79)\\ \text{The researchers are right.}\\ (ii)\\ \text{This is not sensible because in this case the confidence interval}\\ \text{will be too wide and the estimation of population mean will be}\\ \text{too inaccurate.}\\ (iii)\\ \text{They should take the bigger sample or take the smaller confidence level}.\\ 3)\ (i)\\ \text{Population parameters (population means):}\\ a_1=57\text{ --- average lifespan of Ghanaian}\\ (ii)\text{ Sample statistics (sample means):}\\ a_2=55\text{ --- average lifespan of males in Ghana}\\ a_3=60\text{ --- average lifespan of females in Ghana}\\ a_4=58\text{ --- average lifespan of people from the Northern region of Ghana}\\ a_5=54\text{ --- average lifespan of people from the Eastern region of Ghana}\\ (iii)\\ \text{The average lifespan of males is more likely to represent the population mean}\\ \text{(average lifespan of Ghanaian). Because } a_2=55 \text{ is closer to } a_1=57\\ \text{(average lifespan of Ghanaian)}.\\ 4)\ (i)\\ \text{We will use a non-parametric test here because we have no information}\\ \text{about population parameters}.\\ (ii)\\ \text{We will use a parametric test here because we know the population mean}\\ \text{(mean height of all people)}.\\ (iii)\\ \text{We will use a non-parametric test here because we have no information}\\ \text{about population parameters}.\\ (iv)\\ \text{We will use a non-parametric test here because we have no information}\\ \text{about population parameters}.\\ (v)\\ \text{We will use a non-parametric test here because we have no information}\\ \text{about population parameters}.\\$