Let x = age in years of a rural Quebec woman at the time of her first marriage. In the year 1941, the population variance of x was approximately σ² = 5.1. Suppose a recent study of age at first marriage for a random sample of 51 women in rural Quebec gave a sample variance s² = 2.4. Use a 5% level of significance to test the claim that the current variance is less than 5.1. Find a 90% confidence interval for the population variance.

(a) What is the level of significance?


State the null and alternate hypotheses.

H_o: σ² = 5.1; H₁: σ² < 5.1

H_o: σ² = 5.1; H₁: σ² ≠ 5.1    

H_o: σ² < 5.1; H₁: σ² = 5.1

H_o: σ² = 5.1; H₁: σ² > 5.1

(b) Find the value of the chi-square statistic for the sample. (Round your answer to two decimal places.)


What are the degrees of freedom?


What assumptions are you making about the original distribution?

We assume a normal population distribution.

We assume a binomial population distribution.    

We assume a uniform population distribution.

We assume a exponential population distribution.

(c) Find or estimate the P-value of the sample test statistic.

P-value > 0.100 0.050 < P-value < 0.100    

0.025 < P-value < 0.050

0.010 < P-value < 0.025

0.005 < P-value < 0.010

P-value < 0.005

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?

Since the P-value > α, we fail to reject the null hypothesis.

Since the P-value > α, we reject the null hypothesis.    

Since the P-value ≤ α, we reject the null hypothesis.

Since the P-value ≤ α, we fail to reject the null hypothesis.

(e) Interpret your conclusion in the context of the application.

At the 5% level of significance, there is insufficient evidence to conclude that the variance of age at first marriage is less than 5.1.

At the 5% level of significance, there is sufficient evidence to conclude that the variance of age at first marriage is less than 5.1.    

(f) Find the requested confidence interval for the population variance. (Round your answers to two decimal places.)

Interpret the results in the context of the application.

We are 90% confident that σ² lies outside this interval.

We are 90% confident that σ² lies within this interval.    

We are 90% confident that σ² lies above this interval.

We are 90% confident that σ² lies below this interval.

Tire lifetime:the lifetime of a certain type of automobile tire(in thousands of moles) is normally distributed with mean u=41 and standard deviation o=4 What is the probability that a radon chosen tire has lifetime greater than 50 thousand miles? What proportion of tires have lifetime between 36 and 45 thousand miles What proportion of tires have lifetimes less than 46 thousand miles? Round answers at least 4 places

The age distribution of the Canadian population and the age distribution of a random sample of 455 residents in the Indian community of a village are shown below.

Use a 5% level of significance to test the claim that the age distribution of the general Canadian population fits the age distribution of the residents of Red Lake Village.

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: The distributions are the same.
H₁: The distributions are different.

H₀: The distributions are different.
H₁: The distributions are the same.    

H₀: The distributions are different.
H₁: The distributions are different.

H₀: The distributions are the same.
H₁: The distributions are the same.

(b) Find the value of the chi-square statistic for the sample. (Round your answer to three decimal places.)


Are all the expected frequencies greater than 5?

Yes No    

What sampling distribution will you use?

chi-square

binomial    

uniform

normal

Student's t

What are the degrees of freedom?


(c) Estimate the P-value of the sample test statistic.

P-value > 0.100 0.050 < P-value < 0.100    

0.025 < P-value < 0.050

0.010 < P-value < 0.025

0.005 < P-value < 0.010

P-value < 0.005

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis that the population fits the specified distribution of categories?

Since the P-value > α, we fail to reject the null hypothesis.

Since the P-value > α, we reject the null hypothesis.   

Since the P-value ≤ α, we reject the null hypothesis.

Since the P-value ≤ α, we fail to reject the null hypothesis.

(e) Interpret your conclusion in the context of the application.

At the 5% level of significance, the evidence is insufficient to conclude that the village population does not fit the general Canadian population.

At the 5% level of significance, the evidence is sufficient to conclude that the village population does not fit the general Canadian population.    

*1. The national norm for third graders on a standardized test of reading achievement is a mean score of 27 σ 4 . Rachel determines the mean score on this test for a random sample of third graders from her school district. (a) Phrase a question about her population mean that could be answered by testing a hypothesis. (b) Phrase a question for which an estimation approach would be appropriate.

Needing problem 2 answered, but question 1 goes along with the problem.

*2. The results for Rachel’s sample in Problem 1 is X=33.10 (n=36). (a) Calculate σX . (b) Construct the 95% confidence interval for her population mean score. (c) Construct the 99% confidence interval for her population mean score. (d) What generalization is illustrated by a comparison of your answers to Problems 2b and 2c? Explain in precise terms the meaning of the interval you calculated in Problem 2b.

We are interested in understanding the causes of post-traumatic stress disorder (PTSD) among police officers. We draw a sample of 100 officers from an urban police department and collect three pieces of information:

a) Whether they have or are currently experiencing the symptoms of PTSD
b) The number of years they have been on the police force
c) Whether or not they used their firearm to shoot someone during their duties

Construct a 95% confidence interval by hand for what proportion of students are vegan, if you took a random sample of 100 students and 3 were vegan.

The following table shows ceremonial ranking and type of pottery sherd for a random sample of 434 sherds at an archaeological location.

Use a chi-square test to determine if ceremonial ranking and pottery type are independent at the 0.05 level of significance.

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: Ceremonial ranking and pottery type are not independent.
H₁: Ceremonial ranking and pottery type are independent.

H₀: Ceremonial ranking and pottery type are independent.
H₁: Ceremonial ranking and pottery type are not independent.    

H₀: Ceremonial ranking and pottery type are not independent.
H₁: Ceremonial ranking and pottery type are not independent.

H₀: Ceremonial ranking and pottery type are independent.
H₁: Ceremonial ranking and pottery type are independent.

(b) Find the value of the chi-square statistic for the sample. (Round the expected frequencies to at least three decimal places. Round the test statistic to three decimal places.)


Are all the expected frequencies greater than 5?

Yes No    

What sampling distribution will you use?

binomial chi-square     normal Student's t uniform

What are the degrees of freedom?


(c) Find or estimate the P-value of the sample test statistic. (Round your answer to three decimal places.)

p-value > 0.100 0.050 < p-value < 0.100    

0.025 < p-value < 0.050

0.010 < p-value < 0.025

0.005 < p-value < 0.010

p-value < 0.005

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis of independence?

Since the P-value > α, we fail to reject the null hypothesis.

the P-value > α, we reject the null hypothesis.    

Since the P-value ≤ α, we reject the null hypothesis.

Since the P-value ≤ α, we fail to reject the null hypothesis.

(e) Interpret your conclusion in the context of the application.

At the 5% level of significance, there is sufficient evidence to conclude that ceremonial ranking and pottery type are not independent.

At the 5% level of significance, there is insufficient evidence to conclude that ceremonial ranking and pottery type are not independent.    

Follow the instructions carefully. [20 marks] Question Sampling is the process of selecting a representative subset of observations from a population to determine characteristics (i.e. the population parameters) of the random variable under study. Probability sampling includes all selection methods where the observations to be included in a sample have been selected on a purely random basis from the population. Briefly explain FIVE (5) types of probability sampling.

A survey asks a random sample of 1500 adults in Ohio if they support an increase in the state sales tax from 5% to 6%, with the additional revenue going to education. Let P denote the proportion in the sample who say they support the increase. Suppose that 25% of all adults in Ohio support the increase. The standard deviation of the sampling distribution is_____ . Round your answer to four decimal places.

Suppose that 83% of all dialysis patients will survive for at least 5 years. In a simple random sample of 100 new dialysis patients, what is the probability that the proportion surviving for at least five years will exceed 80%, rounded to 5 decimal places? _____

In performing a chi-square test of independence, as the differences between respective observed and expected frequencies _________, the probability of concluding that the row variable is independent of the column variable increases.

Stay the same, decrease, increase, or double

In order to estimate the average electricity usage per month, a sample of 40 residential customers were selected, and the monthly electricity usage was determined using the customers' meter readings. Assume a population variance of 12,100kWh2. Use Excel to find the 98% confidence interval for the mean electricity usage in kilowatt hours. Round your answers to two decimal places and use ascending order. Electric Usage 765 1139 714 687 1027 1109 749 799 911 631 975 717 1232 806 637 894 856 896 1272 1224 621 606 898 723 817 746 933 595 851 1027 770 685 750 1198 975 678 1050 886 826 1176 583 841 1188 692 733 791 584 1163 593 1234 603 1044 1233 1178 598 904 778 693 590 845 893 1028 975 788 1240 1253 854 1185 1164 741 1058 1053 795 1198 1240 1140 959 938 1008 1035 1085 1100 680 1006 977 1042 1252 943 1165 1014 912 791 612 935 864 953 667 1005 1063 1095 1086 810 1032 970 1099 1229 892 1074 579 754 1007 1116 583 763 1231 966 962 1132 738 1033 697 891 840 725 1031

1a) Let an experiment consist of rolling three standard 6-sided dice.

i) Compute the expected value of the sum of the rolls.
ii) Compute the variance of the sum of the rolls.
iii) If X represents the maximum value that appears in the two rolls, what is the expected value of X?

1b) Consider an experiment where a fair die is rolled repeatedly until the first time a 3 is observed.
  
i) What is the sample space for this experiment? What is the probability that the die turns up a 3 after i rolls?
ii) What is the expected number of times we roll the die?
iii) Let E be the event that the first time a 3 turns up is after an even number of rolls. What set of outcomes belong to this event? What is the probability that Eoccurs?

College students are increasingly working while attending school. A study was conducted to determine if there is a significant difference in earnings of male college students and female college students during the eight months of the academic year. A sample of 800 male students earned an average of $5,995 with a standard deviation of $2,500. A sample of 700 working female students earned an average of $5,480 with a standard deviation of $2,600. Use .10 level of significance.

Which of the following represents the null hypothesis?

Around 16% of all ABC college students are declared econ majors. a sample of 75 students is taken.

(1) What is the distribution of the proportion of econ majors in your sample? (sampling distribution)

(2) What is the exact distribution of the number of econ majors in your sample?

(3) What’s the probability that more than a quarter of the students you sample are majoring in econ

(4) What’s the probability that less than 10% of the sample will be econ majors?