1) The PDF of a Gaussian random variable is given by fx(x).

fx(x)= (1/(3*sqrt(2pi) )*e^((x-4)^2)/18

determine

a.) P(X > 4) b). P(X > 0). c). P(X < -2).

2) The joint PDF of random variables X and Y is given by

fxy(x,y)=Ke^-(x+y), x>0 , y>0

Three resistors with resistances R₁, R₂, R₃ are connected in parallel across a battery with voltage V. By Ohm’s law, the current (amps) is

I = V* [ (1/R₁) + (1/R₂) + (1/R₃) ]

Assume that R₁, R₂, R_3, and V are independent random variables

where R₁ ~ Normal (m = 10 ohms, s = 1.5 ohm)

            R₂ ~ Normal (m = 15 ohms, s = 1.5 ohm)

            R₃ ~ Normal (m =20 ohms, s = 1.0 ohms)

            V ~ Normal (m = 120 volts, s = 2.0 volts

(a) Use Monte Carlo Simulation (10,000 random draws from each input random variable) to estimate the mean and standard deviation of the output variable current. (b) Assess whether the output variable current is normally distributed. (c) Assess whether the inverse of current squared (1/ I2 ) is normally distributed. (d) Estimate the probability that the current is less than 25 amps assuming that the inverse of current squared is normally distributed. (e) Compare your answer to (d) with your simulation results – how many of the 10,000 random results for current are below 25 amps via the Stat > Tables > Tally command?

A survey of 2645 consumers by DDB Needham Worldwide of Chicago for public relations agency Porter/Novelli showed that how a company handles a crisis when at fault is one of the top influences in consumer buying decisions,with 73% claiming it is an influence. Quality of product was the number one influence, with 96% of consumers stating that quality influences their buying decisions. How a company handles complaints was number two, with 85% of consumers reporting it as an influence in their buying decisions. Suppose a random sample of 1,100 consumers is taken and each is asked which of these three factors influence their buying decisions.

*a. What is the probability that more than 820 consumers claim that how a company handles a crisis when at fault is an influence in their buying decisions?
**b. What is the probability that fewer than 1,030 consumers claim that quality of product is an influence in their buying decisions?
*c. What is the probability that between 82% and 83% of consumers claim that how a company handles complaints is an influence in their buying decisions?

*(Round the values of z to 2 decimal places. Round the intermediate values to 4 decimal places. Round your answer to 4 decimal places.)
**(Round the values of z to 2 decimal places. Round the intermediate values to 4 decimal places. Round your answer to 5 decimal places.)

A political scientist is interested in the effectiveness of a political ad about a particular issue. The scientist randomly asks 14 individuals walking by to see the ad and then take a quiz on the issue. The general public that knows little to nothing about the issue, on average, scores 51 on the quiz. The individuals that saw the ad scored an average of 45.79 with a variance of 26.01. What can the political scientist conclude with an α of 0.05?

a) What is the appropriate test statistic?
---Select--- na z-test one-sample t-test independent-samples t-test related-samples t-test

b)
Population:
---Select--- general public the ad the particular issue individuals walking by the political ad
Sample:
---Select--- general public the ad the particular issue individuals walking by the political ad

c) Compute the appropriate test statistic(s) to make a decision about H₀.
(Hint: Make sure to write down the null and alternative hypotheses to help solve the problem.)
critical value = ; test statistic =  
Decision:  ---Select--- Reject H0 or  Fail to reject H0

d) If appropriate, compute the CI. If not appropriate, input "na" for both spaces below.
[ , ]

e) Compute the corresponding effect size(s) and indicate magnitude(s).
If not appropriate, input and/or select "na" below.
d =  ;   ---Select--- na trivial effect small effect medium effect large effect
r² =  ;   ---Select--- na trivial effect small effect medium effect large effect

f) Make an interpretation based on the results.

A.Individuals that watched the political ad scored significantly higher on the quiz than the general public.

B.Individuals that watched the political ad scored significantly lower on the quiz than the general public.   

C.Individuals that watched the political ad did not score significantly different on the quiz than the general public.

The contents of bottles of beer are Normally distributed with a mean of 300 ml and a standard deviation of 5 ml.

What is the value in which 90% of the six-packs will have a higher average content?

An exponential probability distribution has a mean equal to 6 minutes per customer. Calculate the following probabilities for the distribution. (PLEASE USE EXCEL FUNCTIONS TO CALCULATE)

A) P(X > 8)

B) P(X > 4)

C) P(6 less than or equal to X less than or equal to 16)

D) P(1 less than or equal to X less than or equal to 5)

The following descriptive statistics are calculated from SPSS. Use them to calculate the test statistic.

1. What is the calculated test statistic?

a) 1.228

b) 3.017

c) 3.249

d) 5.786

2. What is the correct conclusion?

a) Males are not significantly better than females at memory recall (p<.05)

b) Males are significantly better than females at memory recall (p<.05)

c) Males are not significantly better than females at memory recall (p>.05)

d) Males are significantly better than females at memory recall (p>.05)

3. What is the margin of error for calculating a 98% confidence interval for the difference of means?

a) 1.077

b) 2.215

c) 2.479

d) 2.670

4. What are the lower and upper bounds of the 98% confidence interval to estimate the differences of means?

a) 2.423 and 4.577

b) 1.285 and 5.715

c) 1.021 and 5.979

d) 0.830 and 6.170

Find the following probabilities for the standard normal random variable z:

(a) P(−0.76<z<0.75)=

(b) P(−0.98<z<1.36)=

(c) P(z<1.94)=

(d) P(z>−1.2)=

2.

Suppose the scores of students on an exam are Normally distributed with a mean of 480 and a standard deviation of 59. Then approximately 99.7% of the exam scores lie between the numbers ---- and -----. ??

Hint: You do not need to use table E for this problem.

The Aluminum Association reports that the average American uses 56.8 pounds of aluminum in a year. A random sample of 50 households is monitored for one year to determine aluminum usage. If the population standard deviation of annual usage is 12.4 pounds, what is the probability that the sample mean will be each of the following?

a. More than 59 pounds

b. More than 57 pounds

c. Between 56 and 57 pounds

d. Less than 53 pounds

e. Less than 48 pounds

(Round the values of z to 2 decimal places. Round your answers to 4 decimal places.)

Contingency tables may be used to present data representing scales of measurement higher than the nominal scale. For example, a random sample of size 20 was selected from the graduate students who are U.S. citizens, and their grade point averages were recorded. 3.42 3.54 3.21 3.63 3.22 3.8 3.7 3.2 3.75 3.31 3.86 4 2.86 2.92 3.59 2.91 3.77 2.7 3.06 3.3 Also, a random sample of 20 students was selected from the non-U.S. citizen group of graduate students at the same university. Their grade point averages were as follows. 3.50 4.00 3.43 3.85 3.84 3.21 3.58 3.94 3.48 3.76 3.87 2.93 4.00 3.37 3.72 4.00 3.06 3.92 3.72 3.91 Test the null hypothesis that the proportion of graduate students with averages of 3.50 or higher is the same for both the U.S. citizens and the non-U.S. citizens

A carpenter is making doors that are 2058 millimeters tall. If the doors are too long they must be trimmed, and if they are too short they cannot be used. A sample of 5 doors is made, and it is found that they have a mean of 2047 millimeters with a standard deviation of 10 . Is there evidence at the 0.05 level that the doors are too short and unusable? State the null and alternative hypotheses for the above scenario.

A realtor studies the relationship between the size of a house (in square feet) and the property taxes (in $) owed by the owner. The table below shows a portion of the data for 20 homes in a suburb 60 miles outside of New York City. [You may find it useful to reference the t table.] Property Taxes Size 21892 2498 17421 2419 18170 1877 15679 1011 43962 5607 33657 2575 15300 2248 16789 1984 18108 2021 16794 1311 15113 1327 36069 3033 31058 2871 42126 3346 14392 1533 38911 4032 25323 4041 22972 2446 16160 3596 29215 2871

a-1. Calculate the sample correlation coefficient rxy. (Round intermediate calculations to at least 4 decimal places and final answers to 4 decimal places.)

c-1. Calculate the value of the test statistic. (Round intermediate calculations to at least 4 decimal places and final answer to 3 decimal places.)

A political scientist is interested in the effectiveness of a political ad about a particular issue. The scientist randomly asks 18 individuals walking by to see the ad and then take a quiz on the issue. The general public that knows little to nothing about the issue, on average, scores 50 on the quiz. The individuals that saw the ad scored an average of 49.61 with a standard deviation of 5.02. What can the political scientist conclude with an α of 0.01?

a) What is the appropriate test statistic?
---Select--- na z-test one-sample t-test independent-samples t-test related-samples t-test

b)
Population:
---Select--- the political ad general public the particular issue individuals walking by the ad
Sample:
---Select--- the political ad general public the particular issue individuals walking by the ad

c) Compute the appropriate test statistic(s) to make a decision about H₀.
(Hint: Make sure to write down the null and alternative hypotheses to help solve the problem.)
critical value =  ; test statistic =
Decision:  ---Select--- Reject H0 Fail to reject H0

d) If appropriate, compute the CI. If not appropriate, input "na" for both spaces below.
[  ,  ]

e) Compute the corresponding effect size(s) and indicate magnitude(s).
If not appropriate, input and/or select "na" below.
d =  ;   ---Select--- na trivial effect small effect medium effect large effect
r² =  ;   ---Select--- na trivial effect small effect medium effect large effect

f) Make an interpretation based on the results.

Individuals that watched the political ad scored significantly higher on the quiz than the general public.Individuals that watched the political ad scored significantly lower on the quiz than the general public.    Individuals that watched the political ad did not score significantly different on the quiz than the general public.

True or False? A hypothesis test is conducted at to test whether the population correlation coefficient is zero. If the sample size is 25 and the sample correlation coefficient is 0.6, then the critical values of the student t that define the upper and lower tail rejection areas are 2.069 and -2.069, respectively.

The developers of a new online game have determined from preliminary testing that the scores of players on the first level of the game can be modelled satisfactorily by a Normal distribution with a mean of 185 points and a standard deviation of 28 points. They would like to vary the difficulty of the second level in this game, depending on the player’s score in the first level. (a) The developers have decided to provide different versions of the second level for each of the following groups: (i) those whose score on the first level is in the lowest 25% of scores ii) those whose score on the first level is in the middle 50% of scores (iii) those whose score on the first level is in the highest 25% of scores. Use the information given above to determine the cut-off scores for these groups. (You may round each of your answers to the nearest whole number.) (b) In the second level of the game, the developers have also decided to give players an opportunity to qualify for a bonus round. Their stated aim is that players from group (i) should have 75% chance of qualifying for the bonus round, players from group (ii) should have 55% chance of qualifying for the bonus round and that players from group (iii) should have 30% chance of qualifying for this round. Let ?, ? and ? respectively denote the events that a player’s score on the first level was in the lowest 25% of scores, the middle 50% of scores and the highest 25% of scores, and let ? denote the event that the player qualifies for the bonus round. Use event notation to express the developers’ aim as a set of conditional probabilities. (c) Based on the developers’ stated aim, find the total probability that a randomly chosen player will qualify for the bonus round. (d) Given that a player has qualified for the bonus round, what is the probability that the player’s score on the first level was in the middle 50% of scores for that level? (e) Given that a player has not qualified for the bonus round, what is the probability that the player’s score on the first level was in the lowest 25% of scores for that level?