The waiting times X and Y (in minutes) of two clients A and B who are standing in line at two different check outs in the supermarket are modeled as independent, exponential random variables with parameter 1.

(a) Find the cumulative distribution function of the random variable M :=min{X,Y} where min{x,y} is just the smaller value of the two numbers.

(b) Find the probability density function of M. Do you recognize the socalled probability law or probability distribution of the random variable M?

(c) What is the probability that both clients wait more than 2 minutes?

Total plasma volume is important in determining the required plasma component in blood replacement therapy for a person undergoing surgery. Plasma volume is influenced by the overall health and physical activity of an individual. Suppose that a random sample of 42 male firefighters are tested and that they have a plasma volume sample mean of x = 37.5 ml/kg (milliliters plasma per kilogram body weight). Assume that σ = 7.20 ml/kg for the distribution of blood plasma.

(a)

Find a 99% confidence interval for the population mean blood plasma volume in male firefighters. What is the margin of error? (Round your answers to two decimal places.)

lower limitupper limitmargin of error

(b)

What conditions are necessary for your calculations? (Select all that apply.)

the distribution of weights is uniformn is largeσ is unknownσ is knownthe distribution of weights is normal

(c)

Interpret your results in the context of this problem.

The probability that this interval contains the true average blood plasma volume in male firefighters is 0.99.99% of the intervals created using this method will contain the true average blood plasma volume in male firefighters.    1% of the intervals created using this method will contain the true average blood plasma volume in male firefighters.The probability that this interval contains the true average blood plasma volume in male firefighters is 0.01.

(d)

Find the sample size necessary for a 99% confidence level with maximal margin of error E = 2.00 for the mean plasma volume in male firefighters. (Round up to the nearest whole number.)

male firefighters

In this problem, assume that the distribution of differences is approximately normal. Note: For degrees of freedom d.f. not in the Student's t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value by a small amount and therefore produce a slightly more "conservative" answer.

Suppose that at five weather stations on Trail Ridge Road in Rocky Mountain National Park, the peak wind gusts (in miles per hour) for January and April are recorded below.

Wilderness District	1	2	3	4	5
January	127	138	139	64	78
April	107	105	115	88	61

Does this information indicate that the peak wind gusts are higher in January than in April? Use α = 0.01. Solve the problem using the critical region method of testing. (Let d = January − April. Round your answers to three decimal places.)

test statistic	=
critical value	=

Please use excel for test statistic and p value not calculator

Research commissioned by Vodafone suggests that older

workers are the happiest employees (BBC News, July 21,

2008). The report documents that 70% of older workers in

England feel fulfilled, compared with just 50% of younger

workers. A demographer believes that an identical pattern

does not exist in Asia. A survey of 120 older workers in

Asia finds that 75 feel fulfilled. A similar survey finds that

58% of 210 younger workers feel fulfilled.

a- At the 5% level of significance, test if older workers in

Asia feel less fulfilled than their British counterparts.

b-At the 5% level of significance, test if younger workers

in Asia feel more fulfilled than their British

A random sample of 36 individuals were selected from the batch of products to estimate a feature of interest. Suppose that the sample mean is 125 and the standard deviation for this batch is assumed to be 24.

a) construct a 95% confidence interval for the mean value for this feature of the batch.

b) if we want to make the sampling error +/- 7, how many more individuals should be selected to achieve a confidence level 95%?

c) based on the current sample, how confident can you claim that the true value for the mean is between 118 and 132?

Suppose a geyser has a mean time between eruptions of 93 minutes. If the interval of time between the eruptions is normally distributed with standard deviation 28 minutes , answer the following questions.

(a) What is the probability that a randomly selected time interval between eruptions is longer than 107 minutes?

(b) What is the probability that a random sample of 7 time intervals between eruptions has a mean longer than 107 minutes?

(c) What is the probability that a random sample of 21 time intervals between eruptions has a mean longer than 107 minutes?

(d) What effect does increasing the sample size have on the probability? Provide an explanation for this result. Choose the correct answer below.

A. The probability increases because the variability in the sample mean increases as the sample size increases.

B. The probability decreases because the variability in the sample mean increases as the sample size increases.

C. The probability increases because the variability in the sample mean decreases as the sample size increases.

D. The probability decreases because the variability in the sample mean decreases as the sample size increases.

(e) What might you conclude if a random sample of 21 time intervals between eruptions has a mean longer than 107 minutes? Choose the best answer below.

A. The population mean may be greater than 93.The population mean may be greater than 93.

B. The population mean must be more than 93, since the probability is so low. The population mean must be more than 93, since the probability is so low.

C. The population mean must be less than 93, since the probability is so low. The population mean must be less than 93, since the probability is so low.

D. The population mean is 93 minutes, and this is an example of a typical sampling.

Create a simulation layout for rolling 3 dice 627 times. Calculate the probability of the 3 adding to 11.

***Use Excel and show formulas used***

A transect is an archaeological study area that is 1/5 mile wide and 1 mile long. A site in a transect is the location of a significant archaeological find. Let x represent the number of sites per transect. In a section of Chaco Canyon, a large number of transects showed that x has a population variance σ² = 42.3. In a different section of Chaco Canyon, a random sample of 19 transects gave a sample variance s² = 48.5 for the number of sites per transect. Use a 5% level of significance to test the claim that the variance in the new section is greater than 42.3. Find a 95% confidence interval for the population variance.

(a) What is the level of significance?


State the null and alternate hypotheses.

H_o: σ² > 42.3; H₁: σ² = 42.3

H_o: σ² = 42.3; H₁: σ² < 42.3    

H_o: σ² = 42.3; H₁: σ² > 42.3

H_o: σ² = 42.3; H₁: σ² ≠ 42.3

(b) Find the value of the chi-square statistic for the sample. (Round your answer to two decimal places.)


What are the degrees of freedom?


What assumptions are you making about the original distribution?

We assume a normal population distribution.We assume a uniform population distribution.    

We assume a binomial population distribution.We assume a exponential population distribution.

(c) Find or estimate the P-value of the sample test statistic.

P-value > 0.100

0.050 < P-value < 0.100    

0.025 < P-value < 0.050

0.010 < P-value < 0.025

0.005 < P-value < 0.010

P-value < 0.005

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?

Since the P-value > α, we fail to reject the null hypothesis.

Since the P-value > α, we reject the null hypothesis.    

Since the P-value ≤ α, we reject the null hypothesis.

Since the P-value ≤ α, we fail to reject the null hypothesis.

(e) Interpret your conclusion in the context of the application.

At the 5% level of significance, there is insufficient evidence to conclude conclude that the variance is greater in the new section.

At the 5% level of significance, there is sufficient evidence to conclude conclude that the variance is greater in the new section.    

(f) Find the requested confidence interval for the population variance. (Round your answers to two decimal places.)

lower limit
upper limit

Interpret the results in the context of the application.

We are 95% confident that σ² lies below this interval.

We are 95% confident that σ² lies within this interval.    

We are 95% confident that σ² lies outside this interval.

We are 95% confident that σ² lies above this interval.

The continuous random variables ? and ? have the joint p.d.f. ??,? given by

?(?, ?) = { ?^ (−?−?) , ? > 0, ? > 0

0 otherwise

a) Find the conditional p.d.f. ??|?(?|?) of ? given ? = ?. [1]

b) Find ?(?|? = ?) and ???(?|? = ?). [1]

c) Find ?(?) and ???(?). [1]

d) Find ???(?, ?). [1]

Justify the following: Ture or False

1.)A good point estimate is both unbiased and has small standard deviation.

2.)The confidence level C is NOT the probability that the true parameter values is contained in a specified confidence interval.

3.)If you took 100 samples and computed 95% confidence intervals for (u) in each. Approximatley 95% of the confidence intervals will contain the true parameter value.

4.) It is useless to compute a confidence interval using census data.

In answering the question(s), make sure to write down the following 7 steps.

Step 1: Establish null and alternate hypotheses State the null and alternative hypothesis (as a sentence and formula).

Step 2: Calculate the degrees of freedom

Step 3: Calculate t _critical using critical t – table

Step 4: Calculate the Sum of Square deviation (SS_D)

Step 5: Calculate t _obtained

Step 6: Specify the critical value and the obtained value on a t-distribution curve

Step 7: Decision and Conclusion Write a clear and concise conclusion.

A researcher is interested in whether gender predicts the type of costume people would wear to an upcoming party. He asks 6 guys and 6 girls: How likely would you be to wear a “Creepy Clown” costume to the upcoming Halloween party? (1 = very unlikely to 7 = very likely). The data are shown below.

Guys: 4 6 4 3 6 7

Girls: 2 5 4 1 2 3

1. Use alpha = .01 to see whether gender impacts willingness to wear a Creepy Clown costume. (Note: You need to write down all the 7 steps.)

2. Use alpha = .01 to see whether guys are more willing than girls to a Creepy Clown costume. (Note: You only need to write down the steps that are different from part (1).)

3. Use alpha = .05 to see whether guys are more willing than girls to a Creepy Clown costume. (Note: You only need to write down the steps that are different from part (1).)

Develop quantitative null and alternative hypotheses for a decision that is relevant to your life. This can be a personal item or something at work. Be sure to define/discuss the following in your post:

-All variables (include numerical values for your variables)

-The appropriate test statistic, and whether it is a one- or two-tailed test

-Explain your selection process

-Identify the Type I and Type II Errors that could occur with your decision‐making process

-Lastly, share your proposed next steps based on your results

-Be sure to include your references to support your findings

 
Employee
Male
Female
Male
Female
1
20.3
14.3
15.2
19.3
2
19.7
19.5
14.2
18.4
3
27.6
16.5
8.9
18.1
4
25.8
13.8
9.7
15.9
5
26.2
12.4
10.5
17.8
6
20.4
11.8
15.1
17.0
7
24.3
14.2
12.0
16.8
Total
164.3
102.5
85.6
123.3
Mean
23.47
14.64
12.22
17.61
GT
164.3+102.5+85.6+123.3=475.7

1. A reaction time test has a population mean of 150 milliseconds and a population standard deviation of 25 millisenconds. Use this information to complete the problems below:

a. What proportion of reaction times is longer than 140 milliseconds?

b. What proportion of reaction times is shorter than 130 milliseconds?

c. What proportion of reaction times fall between 120 and 145 milliseconds?

d. What reaction time represents the 75th percentile?

e. What reaction time represents the 10th percentile?

f. If you were to guess the suit of a card (hearts, diamonds, spades, clubs), what is the propbability of prediciting the suit correctly in more than 19 trails out of 52 trials?

Are America's top chief executive officers (CEOs) really worth all that money? One way to answer this question is to look at row B, the annual company percentage increase in revenue, versus row A, the CEO's annual percentage salary increase in that same company. Suppose that a random sample of companies yielded the following data:

B: Percent for company 28 16 25 26 18 20 7 10

A: Percent for CEO 23 14 23 18 23 10 4 14

Do these data indicate that the population mean percentage increase in corporate revenue (row B) is different from the population mean percentage increase in CEO salary? Use a 5% level of significance. Find (or estimate) the P-value.

Select one:

a. P-value = 0.50

b. P-value = 0.40

c. 0.02 < P-value < 0.05

d. 0.20 < P-value < 0.40

e. 0.01 < P-value < 0.02