Maths hw

1

Math 140 Exam 2 COC Spring 2022

150 Points

Question 1 (30 points) Match the following vocabulary words in the table below with the corresponding definitions.

Confidence Interval Hypothesis Test Standard Error Alternative Hypothesis

Randomized Simulation Random Sample Random Assignment Random Chance

Population Sampling Variability Significance Level Type II Error

One-Population Mean T-Test Statistic

Quantitative Data One-Population Proportion Z-Test

Statistic

Categorical Data

Critical Value Statistic Parameter Census

Type I Error Bootstrap Distribution Margin of Error Beta Level

Bootstrapping Null Hypothesis P-value Point Estimate

a. A number we compare our test statistic to in order to determine significance. In a sampling

distribution or a theoretical distribution approximating the sampling distribution, the critical

value shows us where the tail or tails are. The test statistic must fall in the tail to be significant.

b. Also called the Alpha Level. If the P-value is lower than this number, then the sample data

significantly disagrees with the null hypothesis and is unlikely to have happened by random

chance. This is also the probability of making a type 1 error.

c. A statement about the population that does not involve equality. It is often a statement about a

“significant difference”, “significant change”, “relationship” or “effect”.

d. The collection of all people or objects you want to study.

e. A number calculated from sample data in order to understand the characteristics of the data.

f. When biased sample data leads you to support the alternative hypothesis when the alternative

hypothesis is actually wrong in the population.

g. Another word for sampling variability. The principle that random samples from the same

population will usually be different and give very different statistics.

h. Data in the form of numbers that measure or count something. They usually have units and

taking an average makes sense.

i. Taking many random samples values from one original real random sample with replacement.

j. Collecting data from everyone in a population.

2

k. Collecting data from a population in such a way that every person in the population has an

approximately equal chance of being chosen. This technique tends to give us data with less

sampling bias.

l. The probability of getting the sample data or more extreme because of sampling variability (by

random chance) if the null hypothesis is true.

m. The sample proportion is this many standard errors above or below the population proportion in

the null hypothesis.

n. Take a group of people or objects and randomly put them into two or more groups. This is a

technique used in experiments to create similar groups. Similar groups help to control

confounding variables so that the scientist can prove cause and effect.

o. Data in the form of labels that tell us something about the people or objects in the data set.

p. The standard deviation of a sampling distribution. The distance that typical sample statistics are

from the center of the sampling distribution. Since the center of the sampling distributions is

usually close to the population parameter, the standard error tells us how far typical sample

statistics are from the population parameter.

q. When someone takes a sample statistic and then claims that it is the population parameter.

r. Two numbers that we think a population parameter is in between. Can be calculated by either a

bootstrap distribution or by adding and subtracting the sample statistic and the margin of error.

s. When biased sample data leads you fail to reject the null hypothesis when the null hypothesis is

actually wrong in the population.

t. The sample mean is this many standard errors above or below the population mean in the null

hypothesis.

u. Putting many bootstrap statistics on the same graph in order to simulate the sampling variability

in a population, calculate standard error, and create a confidence interval. The center of the

bootstrap distribution is the original real sample statistic.

v. The probability of making a type 2 error.

w. A statement about the population that involves equality. It is often a statement about “no

change”, “no relationship” or “no effect”.

x. Random samples values and sample statistics are usually different from each other and usually

different from the population parameter.

y. Total distance that a sample statistic might be from the population parameter. For normal

sampling distributions and a 95% confidence interval, the margin of error is approximately twice

as large as the standard error.

z. A number that describes the characteristics of a population like a population mean or a

population percentage. Can be calculated from an unbiased census, but is often just a guess

about the population.

aa. A procedure for testing a claim about a population.

bb. A technique for visualizing sampling variability in a hypothesis test. The computer assumes the

null hypothesis is true, and then generates random samples. If the sample data or test statistic

falls in the tail, then the sample data significantly disagrees with the null hypothesis. This

technique can also calculate the P-value without a formula.

3

Question 2 (25 Points) 2-Population Mean Hypothesis Test

a. Determine if the following two-population mean tests are matched pair or independent groups.

b. Write the null and alternative hypothesis. Include the claim and what type of test.

c. Check all of the assumptions for a two-population mean T-test. Explain your answers. Does the

problem meets all the assumptions?

d. Write a sentence to explain the T-test statistic.

e. Use the test statistic and the critical value to determine if the sample data significantly disagrees

with the null hypothesis. Explain your answer

f. Write a sentence to explain the P-value.

g. Use the P-value and significance level to determine if the sample data could have occurred by

random chance (sampling variability) or is it unlikely to random chance? Explain your answer.

h. Should we reject the null hypothesis or fail to reject the null hypothesis? Explain your answer.

i. Write a conclusion for the hypothesis test. Explain your conclusion in plain language.

j. Is the categorical variable related to the quantitative variable? Explain your answer.

Here is the scenario.

Researchers collected data on traffic flow, number of shoppers, and traffic accident-related emergency

room admissions on Friday the 13th and the previous Friday, Friday the 6th. They randomly selected 40

Friday the 13ths and 40 Friday the 6ths over a ten-year period. Also given are some sample statistics,

where the difference is the number of cars on the 6th minus the number of cars on the 13th. We claim

that the difference is not zero. (Use a 5% significance level).

Friday 6th Friday 13th Difference

�̅� 128,385 126,550 1835

𝑠 7,664 7,259 405

𝑛 40 40 40

The T-test statistic is ±4.94 and critical value is 2.38. The p-value is < 0.01.

Question 3 (15 Points) 2-Population Mean Confidence Interval

a. Does the data meet the assumptions for inference with two population proportions or two

population means? If it is two means, are the groups independent or matched pair? List the

assumptions needed and how the problem meets them or does not meet them.

b. Does the confidence interval indicate that the mean from population 1 is higher, lower, or not

significantly different from population 2? Explain how you know.

c. Write the two-population confidence interval sentence explaining this confidence interval.

The scenario is the same as in question 2. Also included here is the confidence interval (1518, 2151)

with confidence level 95%.

4

Question 4 (20 Points) 1-Population Proportion Hypothesis Test

a. Give the null and alternative hypothesis. Which is the claim? Is this a right-tailed, left-tailed, or

two-tailed test? Explain how you know what tail to use.

b. Check the assumptions.

c. Give the test statistic and write the standard sentence to explain it. Compare your test statistic

to the critical value. Did the sample data significantly disagree with the null hypothesis? Explain

how.

d. Give the p-value and write the definition sentence to explain it. Could the sample data have

happened because of sampling variability (random chance) or is it unlikely to be sampling

variability? Explain why.

e. Compare the p-value to the significance level. State whether you reject the null hypothesis or

fail to reject the null hypothesis. Explain your answer.

f. Write the standard conclusion.

g. Explain your conclusion in easy to understand language.

Here is the scenario.

An online source suggests that one out of every three people in the U.S have high blood pressure and

the population proportion of U.S. adults is 33.3%. Another website disagrees with this and claims that

the true percentage of U.S. adults with high blood pressure is dramatically lower than 1 in 3 (33.3%). A

random sample of 500 U.S. adults found that 165 of them had high blood pressure. Use the printout and

a 10% significance level to test the claim that less than 33% of U.S. adults have high blood pressure.

N Sample Proportion

Significance Level

Critical Value Test Statistic Z P-value

500 0.33 0.10 -1.282 -0.142 0.4434

Question 5 (10 Points) 1-Population Proportion Bootstrap Confidence Interval

a. Does the data meet the assumptions for a bootstrap confidence interval? Explain your answer.

b. How many bootstrap samples were taken (see picture below)?

c. What is the shape of the bootstrap distribution?

d. Write the upper and lower limits of the bootstrap confidence interval.

e. Write a sentence to explain the bootstrap confidence interval estimate of the population

proportion.

Here is the scenario.

An experiment employing randomization was conducted to see what percentage of rats would show

empathy toward fellow rats in distress. Of the 30 total rats in the study, 23 showed empathy. Use the

bootstrap distribution to find a 99% confidence interval for the population proportion.

5

Question 6 (20 Points) a. State the Central Limit Theorem for Means and explain the ideas behind it.

b. Describe the process of making a sampling distribution.

c. What is a point estimate? Discuss how point estimates create confusion for people reading

articles and scientific reports.

d. What conditions should be met to ensure that a sampling distribution of sample proportions is

normal?

Question 7 (5 Points) State the assumptions for a one-population variance or standard deviation confidence interval.

6

Question 8 (5 Points) Use the following formulas to identify the sample statistic (p-hat or x-bar or s) and the margin of error.

𝑆𝑎𝑚𝑝𝑙𝑒 𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐 = (𝑈𝑝𝑝𝑒𝑟 𝐿𝑖𝑚𝑖𝑡 + 𝐿𝑜𝑤𝑒𝑟 𝐿𝑖𝑚𝑖𝑡)

2

𝑀𝑎𝑟𝑔𝑖𝑛 𝑜𝑓 𝐸𝑟𝑟𝑜𝑟 = (𝑈𝑝𝑝𝑒𝑟 𝐿𝑖𝑚𝑖𝑡 − 𝐿𝑜𝑤𝑒𝑟 𝐿𝑖𝑚𝑖𝑡)

2

A 95% confidence interval estimate of the population proportion of pies in a county fair in Ohio is

(0.46, 0.52).

Question 9 (20 Points) Type I and II Errors

Scenario I: Mike and his advertisement team have created an advertisement plan for a new flavor of soda. Right

now, approximately 16% of soda drinkers are purchasing this flavor. Mike needs to show his bosses that

his advertisement plan will increase the percentage of soda drinkers purchasing this new flavor. If Mike’s

advertising team succeeds in increasing the percentage of customers that prefer this new flavor, then

the company will increase supply and make more of the soda to meet demand. If not, then the company

will keep the supply as it currently is. After the advertising changes, Mike takes a random sample of

customers to determine if the percentage of soda drinkers that like the new flavor has increased. (They

are currently using a 5% significance level).

𝐻0: 𝜋 = 0.16 (𝑇ℎ𝑒 𝑐𝑜𝑚𝑝𝑎𝑛𝑦 𝑤𝑖𝑙𝑙 𝑛𝑜𝑡 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑛𝑒𝑤 𝑓𝑙𝑎𝑣𝑜𝑟 𝑜𝑓 𝑠𝑜𝑑𝑎. )

𝐻𝐴: 𝜋 > 0.16 (𝑇ℎ𝑒 𝑐𝑜𝑚𝑝𝑎𝑛𝑦 𝑤𝑖𝑙𝑙 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑛𝑒𝑤 𝑓𝑙𝑎𝑣𝑜𝑟 𝑜𝑓 𝑠𝑜𝑑𝑎. )

a. Write a description of a type 1 error and the possible consequences of that error in the context

of the problem.

b. Write a description of a type 2 error and possible consequences of that error in the context of

the problem.

c. Would you recommend any changes to the significance level or sample size based on what you

know about the type 1 and type 2 errors in this problem? Explain.

Scenario II: A global sportswear company is contemplating contributing money to a political candidate in the next

election. The managers of the company do not want to contribute unless they are sure the candidate

will get the majority of the population vote and win the election. Otherwise, the company will not

contribute to the candidates’ campaign. (They are currently using a 10% significance level).

𝐻0: 𝜋 ≤ 0.5 (𝑇ℎ𝑒 𝑐𝑜𝑚𝑝𝑎𝑛𝑦 𝑤𝑖𝑙𝑙 𝑛𝑜𝑡 𝑐𝑜𝑛𝑡𝑟𝑖𝑏𝑢𝑡𝑒 𝑡𝑜 𝑡ℎ𝑒 𝑝𝑜𝑙𝑖𝑡𝑖𝑐𝑎𝑙 𝑐𝑎𝑛𝑑𝑖𝑑𝑎𝑡𝑒𝑠 ′ 𝑐𝑎𝑚𝑝𝑎𝑖𝑔𝑛. )

𝐻𝐴: 𝜋 > 0.5 (𝑇ℎ𝑒 𝑐𝑜𝑚𝑝𝑎𝑛𝑦 𝑤𝑖𝑙𝑙 𝑐𝑜𝑛𝑡𝑟𝑖𝑏𝑢𝑡𝑒 𝑡𝑜 𝑡ℎ𝑒 𝑝𝑜𝑙𝑖𝑡𝑖𝑐𝑎𝑙 𝑐𝑎𝑛𝑑𝑖𝑑𝑎𝑡𝑒𝑠 ′ 𝑐𝑎𝑚𝑝𝑎𝑖𝑔𝑛. )

7

a. Write a description of a type 1 error and the possible consequences of that error in the context

of the problem.

b. Write a description of a type 2 error and possible consequences of that error in the context of

the problem.

c. Would you recommend any changes to the significance level or sample size based on what you

know about the type 1 and type 2 errors in this problem? Explain.