Because of safety considerations, in May 2003 the Federal Aviation Administration (FAA) changed its guidelines for how small commuter airlines must estimate passenger weights. Under the old rule, airlines used 180 pounds as a typical passenger weight (including carry-on luggage) in warm months and 185 pounds as a typical weight in cold months.

A journal reported that an airline conducted a study to estimate average passenger plus carry-on weights. They found an average summer weight of 183 pounds and a winter average of 190 pounds. Suppose that each of these estimates was based on a random sample of 100 passengers and that the sample standard deviations were 18 pounds for the summer weights and 25 pounds for the winter weights.

(a)

Construct a 95% confidence interval for the mean summer weight (including carry-on luggage) of this airline's passengers. (Round your answers to three decimal places.)

Interpret a 95% confidence interval for the mean summer weight (including carry-on luggage) of this airline's passengers.

There is a 95% chance that the true mean summer weight (including carry-on luggage) of this airline's passengers is one of these two values. There is a 95% chance that the true mean summer weight (including carry-on luggage) of this airline's passengers is directly in the middle of these two values.     We are 95% confident that the true mean summer weight (including carry-on luggage) of this airline's passengers is between these two values. There is a 95% chance that the true mean summer weight (including carry-on luggage) of this airline's passengers is between these two values. We are 95% confident that the true mean summer weight (including carry-on luggage) of this airline's passengers is directly in the middle of these two values.

(b)

Construct a 95% confidence interval for the mean winter weight (including carry-on luggage) of this airline's passengers. (Round your answers to three decimal places.)

Interpret a 95% confidence interval for the mean winter weight (including carry-on luggage) of this airline's passengers.

There is a 95% chance that the true mean winter weight (including carry-on luggage) of this airline's passengers is one of these two values. There is a 95% chance that the true mean winter weight (including carry-on luggage) of this airline's passengers is between these two values.     We are 95% confident that the true mean winter weight (including carry-on luggage) of this airline's passengers is between these two values. There is a 95% chance that the true mean winter weight (including carry-on luggage) of this airline's passengers is directly in the middle of these two values. We are 95% confident that the true mean winter weight (including carry-on luggage) of this airline's passengers is directly in the middle of these two values.

(c)

The new FAA recommendations are 190 pounds for summer and 195 pounds for winter. Comment on these recommendations in light of the confidence interval estimates from part (a) and part (b).

Only the new winter FAA recommendation seems accurate, since only the new winter recommendation value is contained within its 95% confidence interval. Only the new summer FAA recommendation seems accurate, since only the new summer recommendation value is contained within its 95% confidence interval.     These new FAA recommendations don't seem accurate, since neither recommendation value is contained within its respective 95% confidence interval. These new FAA recommendations seem accurate, since both recommendation values are contained within their respective 95% confidence interval.

You may need to use the appropriate table in Appendix A to answer this question.

The number of undergraduate students at the University of Winnipeg is approximately 9,000, while the University of Manitoba has approximately 27,000 undergraduate students. Suppose that, at each university, a simple random sample of 3% of the undergraduate students is selected and the following question is asked: “Do you approve of the provincial government’s decision to lift the tuition freeze?”. Suppose that, within each university, approximately 20% of undergraduate students favour this decision. What can be said about the sampling variability associated with the two sample proportions?
(A) The sample proportion for the U of W has less sampling variability than that for the U of M.

(B) The sample proportion for the U of W has more sampling variability that that for the U of M.

(C) The sample proportion for the U of W has approximately the same sampling variability as that for the U of M.

(D) It is impossible to make any statements about the sampling variability of the two sample proportions without taking many samples.

(E) It is impossible to make any statements about the sampling variability of the two sample proportions because the population sizes are different.

Could you explain why answer is (B)

What business, economic, policy, and environmental threats to organization growth are CEOs extremely concerned about? In a survey by PricewaterhouseCoopers (PwC), 57 of 114 U.S. CEOs are extremely concerned about cyber threats, and 22 are extremely concerned about lack of trust in business.

a. Construct a 95% confidence interval estimate for the population proportion of U.S. CEOs who are extremely concerned about cyber threats.

b. Construct a 95% confidence interval estimate for the population proportion of U.S. CEOs who are extremely concerned about lack of trust in business.

c. Interpret the intervals in (a) and (b).

let v and u are random variables fvu(v,u)= e^-v-u for u ,v>=0 Z= e^v+u find fz(z) ?

Company XYZ know that replacement times for the quartz time pieces it produces are normally distributed with a mean of 15.7 years and a standard deviation of 1.2 years.

Find the probability that a randomly selected quartz time piece will have a replacement time less than 13.4 years?
P(X < 13.4 years) =

Enter your answer accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

If the company wants to provide a warranty so that only 1.5% of the quartz time pieces will be replaced before the warranty expires, what is the time length of the warranty?
warranty =   years

Enter your answer as a number accurate to 1 decimal place. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

A certain flight arrives on time

8383

percent of the time. Suppose

147147

flights are randomly selected. Use the normal approximation to the binomial to approximate the probability that ​(a) exactly

128128

flights are on time.​(b) at least

128128

flights are on time.​(c) fewer than

127127

flights are on time.​(d) between

127127

and

132132​,

inclusive are on time.

A company that produces coffee for use in commercial machines monitors the caffeine content in its coffee. The company selects 35 8-oz samples each hour from its production line to analyze. The samples collected one morning between 8:00 - 9:00 am contained on average 96.1 mg of caffeine, with standard deviation 1.2 mg.

1. a) Compute and interpret a 95% confidence interval for mean caffeine content based on the collected data.

2. b) According to production standards, the mean amount of caffeine content per 8 ounces should be no more than 95 mg. An overly high caffeine content indicates that the coffee beans have not been roasted long enough.

   Conduct a formal hypothesis test to investigate whether production standards are being met, based on the observed data. Summarize your findings to the CEO using language accessible to someone who has not taken a statistics course and make a recommendation as to whether an adjustment needs to be made to the bean roasting time.

3. c) A set of samples collected between 10:00 - 11:00 am on the same day has average caffeine content of 95.3 mg, with standard deviation 1.1 mg. Based on observing this data, would you change your recommendation in part c)? Explain your answer.

(A derivation of the bivariate normal distribution) let $Z_{1}$ and $Z_{2}$ be independent n(0,1) random variables, and define new random variables X and Y by\\
\begin{align*}
X=a_{x}Z_{1}+b_{x}Z_{2}+c_{x} \quad Y=a_{Y}Z_{1}+b_{Y}Z_{2}+c_{Y}\\
\end{align*}
where $a_{x},b_{x},c_{x},a_{Y},b_{Y},c_{Y}$ are constants.\\
if we define the constants $a_{x},b_{x},c_{x},a_{Y},b_{Y},c_{Y}$ by\\
\begin{align*}
a_{x}=\sqrt{\frac{1+\rho}{2}}\sigma_{X}, b_{x}=\sqrt{\frac{1-\rho}{2}}\sigma_{X}, c_{X}=\mu_{X},\\
a_{Y}=\sqrt{\frac{1+\rho}{2}}\sigma_{Y}, b_{Y}=-\sqrt{\frac{1-\rho}{2}}\sigma_{Y},c_{Y}=\mu_{Y}\\
\end{align*}
where \mu_{X}, \mu_{Y},\sigma^2_{x},\sigma^2_{Y},\rho are constants\\
\\
\begin{itemize}
\item
Question 1):\\
Show that $(X,Y)$ has the bivariate normal pdf with parameter\\s $\mu_{X}, \mu_{Y},\sigma^2_{x},\sigma^2_{Y},\rho$\\
\item
Question 2):\\
if we start with bivariate normal parameters $$\mu_{X}, \mu_{Y},\sigma^2_{x},\sigma^2_{Y},\rho$, define constants $a_{x},b_{x},c_{x},a_{Y},b_{Y},c_{Y}$ as the solutions to the equations:\\
\begin{align*}
\mu_{X}=c_{X}, \sigma^2_{X}=a^2_{X}+b^2_{X},\\
\mu_{Y}=c_{Y}, \sigma^2_{Y}=a^2_{Y}+b^2_{Y},\\
\rho\sigma_{X}\sigma_{Y}=a_{X}a_{Y}+b_{X}b_{Y}
\end{align*}
  
\end{itemize}

\end{document}

A food safety guideline is that the mercury in fish should be below 1 part per million​ (ppm). Listed below are the amounts of mercury​ (ppm) found in tuna sushi sampled at different stores in a major city. Construct a 90​% confidence interval estimate of the mean amount of mercury in the population. Does it appear that there is too much mercury in tuna​ sushi? 0.60  0.69  0.09  0.94  1.31  0.56  0.92

What is the confidence interval estimate of the population mean mu​?

The National Football League (NFL) records a variety of performance data for individuals and teams. To investigate the importance of passing on the percentage of games won by a team, the following data show the conference (Conf), average number of passing yards per attempt (Yds/Att), the number of interceptions thrown per attempt (Int/Att), and the percentage of games won (Win%) for a random sample of  NFL teams for a season

a. Develop the estimated regression equation that could be used to predict the percentage of games won given the average number of passing yards per attempt (to 1 decimal). Enter negative value as negative number.

WIN%=______+_______*Yds/Att

b. Develop the estimated regression equation that could be used to predict the percentage of games won given the number of interceptions thrown per attempt (to 1 decimal). Enter negative value as negative number.

      WIN%=______+_______*INT/Att

c. Develop the estimated regression equation that could be used to predict the percentage of games won given the average number of passing yards per attempt and the number of interceptions thrown per attempt (to 1 decimal). Enter negative value as negative number.

        WIN%=______+_______*Yds/Att+______*INT/Att

d. The average number of passing yards per attempt for the Kansas City Chiefs was 6.2 and the number of interceptions thrown per attempt was 0.036 . Use the estimated regression equation developed in part (c) to predict the percentage of games won by the Kansas City Chiefs. (Note: For a season the Kansas City Chiefs' record was 7 wins and 9 losses.) Compare your prediction to the actual percentage of games won by the Kansas City Chiefs (to whole number).

A toy manufacturer wants to know how many new toys children buy each year. A sample of 292 children was taken to study their purchasing habits. Construct the 80% confidence interval for the mean number of toys purchased each year if the sample mean was found to be 5.6 Assume that the population standard deviation is 1.8. Round your answers to one decimal place.

(a) A random sample of 10 houses in a particular area, each of which is heated with natural gas, is selected and the amount of gas (therms) used during the month of January is determined for each house. The resulting observations are 118, 122, 146, 80, 141, 103, 138, 99, 109, 125. Let μ denote the average gas usage during January by all houses in this area. Compute a point estimate of μ.
therms

(b) Suppose there are 20,000 houses in this area that use natural gas for heating. Let τ denote the total amount of gas used by all of these houses during January. Estimate τ using the data of part (a).
therms

(c) Use the data in part (a) to estimate p, the proportion of all houses that used at least 100 therms.


(d) Give a point estimate of the population median usage (the middle value in the population of all houses) based on the sample of part (a).
therms

Research indicates that the color red increases men’s attraction to women (Elliot & Niesta, 2008). In the original study, men were shown women’s photographs presented on either a white or red background. Photographs presented on red were rated significantly more attractive than the same photographs mounted on white. In a similar study, a researcher prepares a set of 30 women’s photographs, with 15 mounted on a white background and 15 mounted on red. One picture is identified as the test photograph and appears twice in the set, once on white and once on red. Each male participant looks through the entire set of photographs and rates the attractiveness of each woman on a 10-point scale. The following table summarizes the ratings of the test photograph for a sample of nine men. Are the ratings for the test photograph significantly different when it is presented of a red background compared to a white background?

1. State the null and alternative hypotheses
2. Compute the test statistic using SPSS
3. What is the effect size (you will need to compute both Cohen’s d from data given in the SPSS output table)

Cohen’s d= M1 – M2/ ((SD1+SD2) /2)

4. Write the Results section for your findings. Include the descriptive statistics, type of statistical test and results of the test and the confidence interval.

The white wood material used for the roof of an ancient temple is imported from a certain country. The wooden roof must withstand as much as 100 centimeters of snow in the winter. Architects at a university conducted a study to estimate the mean bending strength of the white wood roof. A sample of 16 pieces of the imported wood were tested and yielded the statistics x̄ = 74.6 and s = 9.8 on breaking strength (MPa). Estimate the true mean breaking strength of the white wood with a 95% confidence interval. Interpret the result.

Find the 95% confidence interval for the true mean breaking strength of the white wood.

(___,___) (round to 2 decimal places)

A study of college football games shows that the number of holding penalties assessed has a mean of 2.2 penalties per game and a standard deviation of 1.05 penalties per game. What is the probability that, for a sample of 40 college games to be played next week, the mean number of holding penalties will be 2.15 penalties per game or less?

Carry your intermediate computations to at least four decimal places. Round your answer to at least three decimal places