# MATHEMATICS

**PLEASE READ THE QUESTIONS/PROBLEMS CAREFULLY; SHOW ALL YOUR WORK AND REASONING, ***Just the answer, without supporting work, will receive no credit.*

- Considering the definition of “point estimate,” used in determination of an “interval estimate” (construction of a confidence interval) for an unknown population parameter, determine just the numerical value of the point estimate for a sample of your choice when

- an “interval estimate” for the unknown population mean is of interest.
- an “interval estimate” for the unknown population proportion is of interest.

**NOTE:** You may consider either one sample or two different samples with the size(s) of your choice. For the proportion case you may consider gender, if people are involved in your sample; or, you may consider “defective” and “non-defective” if members of your sample are, say, calculators.

- When you construct a 95% confidence interval, what are you 95% confident about?
- All other things remaining the same, is a 90% confidence interval narrower or wider than a 95% confidence interval? Briefly and clearly explain your answer.
**HINT:**Think of the corresponding margins of error (or error bounds). - All other things remaining the same, what is the impact of an increase in sample size on the width of the confidence interval? Explain your answer briefly and clearly.
- Briefly,

- compare the Z-distribution with Student’s t-distribution (t-distribution), focusing on the four attributes of probability distributions (random variable name, behavior [curve], mean, and standard deviation).
- compare the format of the Z-table with that of the t-table, focusing on the fundamental differences in format and the information involved.

- Briefly answer the following questions:

- When should we use the t-table, instead of the Z-table, while constructing confidence intervals or conducting tests of hypotheses.
- When must the requirement of the normality (or approximate normality) of the population distribution be met when we use Z or t?
- What would be the impact on the confidence interval, if a mistake is made and Z table is used instead of the t-table? Support your answer with a brief description and/or computation.
**HINT:**Examine the superimposed graphs of Z-distribution and t-distribution given in Lane, and focus on the relative magnitudes of Z_{a}_{/2 }and t_{a}_{/2}, each being a factor in the margin of error (error bound).

- (Show your work.) In a recent Zogby International Poll, nine of 48 respondents rated the likelihood of a terrorist attack in their community as “likely” or “very likely.”

- Use the “plus four” method, as explained by Illowsky, to create a 97% confidence interval for the proportion of American adults who believe that a terrorist attack in their community is likely or very likely.
- Explain what this confidence interval means in the context of the problem.

You can use the **statistical tables provided on LEO (Content > Course Resources > Statistical Resources).**

- (Show your work.) A Nissan Motor Corporation advertisement read: “The average man’s I.Q. is 107. The average brown trout’s I.Q. is 4. So why can’t man catch brown trout?” Suppose you believe that the brown trout’s mean I.Q. is greater than four. You catch 12 brown trout. A fish psychologist determines the I.Q.s as follows: 5; 4; 7; 3; 6; 4; 5; 3; 6; 3; 8; 5. Conduct a hypothesis test of your belief. Use a significant level of 5%. The standard deviation of the I.Q.s listed is approximately 1.62. You can use the
**statistical tables provided on LEO (Content > Course Resources > Statistical Resources).** - (Show your work.) According to the Center for Disease Control website, in 2011 at least 18% of high school students have smoked a cigarette. An Introduction to Statistics class in Davies County, KY conducted a hypothesis test at the local high school (a medium sized–approximately 1,200 students–small city demographic) to determine if the local high school’s percentage was lower. One hundred fifty students were chosen at random and surveyed. Of the 150 students surveyed, 82 have smoked. Use a significance level of 0.05 and using appropriate statistical evidence,
**conduct a hypothesis test and state the conclusions.**You can use the**statistical tables provided on LEO (Content > Course Resources > Statistical Resources).** - (Show your work.) A study is done to determine if students in the California state university system take longer to graduate, on average, than students enrolled in private universities. One hundred students from both the California state university system and private universities are surveyed. Suppose that from years of research, it is known that the population standard deviations are 1.5811 years and 1 year, respectively. The following data are collected. The California state university system students took on average 4.5 years with a standard deviation of 0.8. The private university students took on average 4.1 years with a standard deviation of 0.3.
**What would be your conclusions based on significance levels of 5% and 1%?**You can use the**statistical tables provided on LEO (Content > Course Resources > Statistical Resources).**