Q1: In a random sample of70professional actors, it was found that37were extroverts.

(a) Letprepresent the proportion of all actors who are extroverts. Find a point estimate forp. (Round your answer to four decimal places.)

(b) Find a 95% confidence interval forp. (Round your answers to two decimal places.)

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Give a brief interpretation of the meaning of the confidence interval you have found.

We are 5% confident that the true proportion of actors who are extroverts falls within this interval.

We are 5% confident that the true proportion of actors who are extroverts falls above this interval.

We are 95% confident that the true proportion of actors who are extroverts falls within this interval.

We are 95% confident that the true proportion of actors who are extroverts falls outside this interval.

(c) Do you think the conditionsnp>5 andnq>5 are satisfied in this problem? Explain why this would be an important consideration.

No, the conditions are not satisfied. This is important because it allows us to say thatpis approximately binomial.

Yes, the conditions are satisfied. This is important because it allows us to say thatpis approximately binomial.

Yes, the conditions are satisfied. This is important because it allows us to say thatpis approximately normal.

No, the conditions are not satisfied. This is important because it allows us to say thatpis approximately normal.

Q2

In a random sample of530judges, it was found that285were introverts.

(a)

Letprepresent the proportion of all judges who are introverts. Find a point estimate forp. (Round your answer to four decimal places.)

(b)

Find a 99% confidence interval forp. (Round your answers to two decimal places.)

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Give a brief interpretation of the meaning of the confidence interval you have found.

We are 1% confident that the true proportion of judges who are introverts falls above this interval.

We are 99% confident that the true proportion of judges who are introverts falls outside this interval.

We are 99% confident that the true proportion of judges who are introverts falls within this interval.

We are 1% confident that the true proportion of judges who are introverts falls within this interval.

(c)

Do you think the conditionsnp>5 andnq>5 are satisfied in this problem? Explain why this would be an important consideration.

Yes, the conditions are satisfied. This is important because it allows us to say thatpis approximately normal.

No, the conditions are not satisfied. This is important because it allows us to say thatpis approximately binomial.

Yes, the conditions are satisfied. This is important because it allows us to say thatpis approximately binomial.

No, the conditions are not satisfied. This is important because it allows us to say thatpis approximately normal.

Q3

Independent random samples of professional football and basketball players gave the following information. Assume that the weight distributions are mound-shaped and symmetric.

Weights (in lb) of pro football players:x₁;n₁= 21

249 261 254 251 244 276 240 265 257 252

282 256 250 264 270 275 245 275 253 265 271

Weights (in lb) of pro basketball players:x₂;n₂= 19

202 200 220 210 191 215 223 216 228 207

225 208 195 191 207 196 182 193 201

(a) Use a calculator with mean and standard deviation keys to calculatex₁,s₁,x₂, ands₂. (Round your answers to four decimal places.)x₁=s₁=x₂=s₂=

(b) Let₁be the population mean forx₁and let₂be the population mean forx₂. Find a 99% confidence interval for₁₂.(Round your answers to one decimal place.)

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(c) Examine the confidence interval and explain what it means in the context of this problem. Does the interval consist of numbers that are all positive? all negative? of different signs? At the 99% level of confidence, do professional football players tend to have a higher population mean weight than professional basketball players?

Because the interval contains only negative numbers, we can say that professional football players have a lower mean weight than professional basketball players.

Because the interval contains both positive and negative numbers, we cannot say that professional football players have a higher mean weight than professional basketball players.

Because the interval contains only positive numbers, we can say that professional football players have a higher mean weight than professional basketball players.

(d) Which distribution did you use? Why?

The standard normal distribution was used because₁and₂are unknown.

The Student'st-distribution was used because₁and₂are known.

The Student'st-distribution was used because₁and₂are unknown.

The standard normal distribution was used because₁and₂are known.

Q4

A random sample of388married couples found that280had two or more personality preferences in common. In another random sample of574married couples, it was found that only20had no preferences in common. Letp₁be the population proportion of all married couples who have two or more personality preferences in common. Letp₂be the population proportion of all married couples who have no personality preferences in common.

(a) Find a90%confidence interval forp₁-p₂. (Use 3 decimal places.)

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(b) Explain the meaning of the confidence interval in part (a) in the context of this problem. Does the confidence interval contain all positive, all negative, or both positive and negative numbers? What does this tell you (at the90%confidence level) about the proportion of married couples with two or more personality preferences in common compared with the proportion of married couples sharing no personality preferences in common?

Because the interval contains only negative numbers, we can say that a higher proportion of married couples have no personality preferences in common.

We can not make any conclusions using this confidence interval.

Because the interval contains only positive numbers, we can say that a higher proportion of married couples have two or more personality preferences in common.

Because the interval contains both positive and negative numbers, we can not say that a higher proportion of married couples have two or more personality preferences in common.

Q5

The U.S. Geological Survey compiled historical data about Old Faithful Geyser (Yellowstone National Park) from 1870 to 1987. Letx₁be a random variable that represents the time interval (in minutes) between Old Faithful eruptions for the years 1948 to 1952. Based on9380observations, the sample mean interval wasx₁=62.8minutes. Letx₂be a random variable that represents the time interval in minutes between Old Faithful eruptions for the years 1983 to 1987. Based on24,170observations, the sample mean time interval wasx₂=70.8minutes. Historical data suggest that₁=8.42minutes and₂=11.99minutes. Let₁be the population mean ofx₁and let₂be the population mean ofx₂.

(a) Compute a95%confidence interval for₁-₂. (Use 2 decimal places.)

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(b) Comment on the meaning of the confidence interval in the context of this problem. Does the interval consist of positive numbers only? negative numbers only? a mix of positive and negative numbers? Does it appear (at the95%confidence level) that a change in the interval length between eruptions has occurred? Many geologic experts believe that the distribution of eruption times of Old Faithful changed after the major earthquake that occurred in 1959.

Because the interval contains only positive numbers, we can say that the interval length between eruptions has gotten shorter.

Because the interval contains both positive and negative numbers, we can not say that the interval length between eruptions has gotten longer.

We can not make any conclusions using this confidence interval.

Because the interval contains only negative numbers, we can say that the interval length between eruptions has gotten longer.