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

A “subliminal” message is below our threshold of awareness but may nonetheless influence us. A study looked at the effect of subliminal messages on math skills. Messages were flashed on a screen too rapidly to be consciously read. Twenty-eight students who had failed the mathematics part of the City University of New York Skills Assessment Test were randomly assigned to receive daily either a positive subliminal message (“Each day I am getting better in math”) or a neutral subliminal message (“People are walking on the street”). All students took the assessment test again at the end of the program, and the table below gives the data on each of the subjects’ scores before and after the program. Is there statistical evidence that the positive message brought about a greater improvement in math scores than the neutral message? Make no assumptions and show all work.

Got the data by Excel

Naive Bayes Theorem

See the dataset D in Table 1. It consists of clinical data about 14 patients. Using the data in D, determine the Naive Bayes classifier and predict the patients in Table 2. Then, compare with your ‘predicted’ ones with the ground-truth label (i.e., column ’Disease’) and report the accuracy P.

Table 1: Dataset D with clinical data of 14 patients

Table 2: Test data with additional 5 patients

In a developing country, 19% of the entire population has high
speed access to the internet. Random samples of size 200 are selected from the country’s
population in order to estimate the proportion of the population with high-speed internet
access.
a) What is the expected sample proportion? (Do not use the CLT.)
b) What is the standard deviation of the sample proportion? (Do not use the CLT.)
c) What is the probability that the sample proportion is (strictly) less than 14%? (Do not
use the CLT.)
d) Use the Central Limit Theorem (CLT) to estimate the probability that the sample
proportion is between 9% and 29%?
e) Use the CLT to estimate the value p such that the probability that the sample propor
tion is above p is 90%.

Note : Please explain C in details

It is estimated that approximately 8.17% Americans are afflicted with diabetes. Suppose that a certain diagnostic evaluation for diabetes will correctly diagnose 98% of all adults over 40 with diabetes as having the disease and incorrectly diagnoses 3% of all adults over 40 without diabetes as having the disease. a) Find the probability that a randomly selected adult over 40 does not have diabetes, and is diagnosed as having diabetes (such diagnoses are called "false positives"). b) Find the probability that a randomly selected adult of 40 is diagnosed as not having diabetes. c) Find the probability that a randomly selected adult over 40 actually has diabetes, given that he/she is diagnosed as not having diabetes (such diagnoses are called "false negatives"). (Note: it will be helpful to first draw an appropriate tree diagram modeling the situation)