In the following table, the random variable x represents the number of laptop computers that failed during a drop-test of six sample laptops. Use the table to answer the questions a) - e) below.
x 0 1 2 3 4 5 6

P(x) 0.377 0.399 0.176 0.041 0.005 0.000 0.000

a) Find and report the mean and the standard deviation of this distribution.

b) Using the range rule of thumb, identify the range of values containing the usual number of laptop failures among the six laptops that were tested. Is three laptops an unusually high number of failures among six tested? Explain.

c) Find the probability of getting exactly one laptop that fails among six laptops tested.

d) Find the probability of getting one or fewer laptops that fail among six laptops tested.

e) Which probability is most relevant for determining whether one laptop is an unusually low number of laptops to fail among six laptops tested: the result from part (c) or (d)? f) Is one laptop an unusually low number of laptops that fail among six laptops tested? Why or why not?

1) In a recent poll, the Gallup organization found that 45% of adult Americans believe that the overall state of moral values in the United States is poor. Suppose a survey of a random sample of 25 adult Americans is conducted in which they are asked to disclose their feelings on the overall state of moral values in the United States. Answer the questions below, showing work. Bare answers are not acceptable. (Showing work means writing the calculator command you used with correct input values in the correct order.)

f) Would it be unusual to find 20 or more adult Americans who believe the overall state of moral values in the United States is poor? Why or why not?

g) Now based on a random sample of 500 adult Americans, compute the mean and standard deviation of the random variable X, the number of adults who believe the overall state of moral values in the United States is poor. Interpret the mean.

h) Would it be unusual to identify 240 adult Americans who believe the overall state of moral values in the United States is poor, based on a random sample of 500 adult Americans? Why? Now using the Normal Approximation to the Binomial, approximate the probability that:

i) Exactly 250 of those surveyed feel the overall state of moral values in the United States is poor.

j) Between 220 and 250, inclusive, of those surveyed feel the overall state of moral values in the United States is poor.

k) At least 260 adult Americans who believe the overall state of moral values in the United States is poor, based on a random sample of 500 adult Americans. Would this be unusual? Why?

Bayus (1991) studied the mean numbers of auto dealers visited by early and late replacement buyers. Letting μ be the mean number of dealers visited by all late replacement buyers, set up the null and alternative hypotheses needed if we wish to attempt to provide evidence that μ differs from 4 dealers. A random sample of 100 late replacement buyers yields a mean and a standard deviation of the number of dealers visited of x⎯⎯x¯ = 4.26 and s = .52. Using a critical value and assuming approximate normality to test the hypotheses you set up by setting α equal to .10, .05, .01, and .001. Do we estimate that μ is less than 4 or greater than 4? (Round your answers to 3 decimal places.)

H₀ : μ (Click to select)=≠ 4 versus H_a : μ (Click to select)≠= 4.

t         

There is (Click to select)noextremely strongvery strongstrongweak evidence.

μ is (Click to select)less thangreater than 4.

Suppose a random sample of 10,000 individuals is asked to identify their favorite brand of soap among ten choices. The following results from the survey are obtained:

Observed
Brand Frequency

A             1200
B             900
C             850
D             1160
E                1020
F               975
G            1100
H            980
I              1035
J               780

Test the hypothesis that the preferences for each brand are equal (or uniform). Test this at the 0.05 level.

For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

In a random sample of 63 professional actors, it was found that 41 were extroverts.

(a) Let p represent the proportion of all actors who are extroverts. Find point estimates for p and q. (Round your answer to four decimal places.)

p̂=

q̂=

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

Find the maximal margin of error. (Round your answer to two decimal places.)

E =

Report the bounds from the 95% confidence interval for p. (Round your answers to two decimal places.)

lower limit =

upper limit =

We need to find the confidence interval for the SHOE SIZE variable. To do this, we need to find the mean and standard deviation with the Week 1 spreadsheet. Then we can the Week 5 spreadsheet to find the confidence interval. This does not need to be separated by males and females, rather one interval for the entire data set.

First, find the mean and standard deviation by copying the SHOE SIZE variable and pasting it into the Week 1 spreadsheet. Write down the mean and the sample standard deviation as well as the count. Open the Week 5 spreadsheet and type in the values needed in the green cells at the top to find the confidence interval.

1. Give and interpret the 95% confidence interval for the size of students’ shoes.

Change the confidence level to 99% to find the 99% confidence interval for the SHOE SIZE variable.

2. Give and interpret the 99% confidence interval for the size of students’ shoes.

3. Compare the 95% and 99% confidence intervals for the size of students’ shoes. Explain the difference between these intervals and why this difference occurs.

1a)Explain when you can approximate a hypogeometric distribution using a binomial distribution. Why can we do this? Use an example to illustrate the approximation.

b) Prove that if X follows a uniform distribution, the expectation is the average of all the outcomes.

1. How many scores are between 2 standard deviations above and below the mean in a normal distribution?
2. A score is 3 standard deviations above the mean in a normal distribution. How much of the data is below that score?
3. How many combinations can nine pair of shoes be made for nine people?

For the following questions, find the probability using a standard 52-card deck. Write your answer as a fraction or with a colon in lowest terms.

1. Find the probability of drawing a face card.
2. Find the probability of drawing a black card.
3. Find the probability of drawing a red 10.
4. Find the probability of drawing a king.
5. Find the probability of not drawing a face card.
6. Find the probability of drawing a number less than 4.
7. Find the probability of drawing a 5.
8. Find the probability of drawing a black 2.
9. Find the probability of not drawing a red 8.
10. Find the probability of drawing an ace of spades.

Complete the frequency table above by filling in the frequency of raw scores occurring in each interval, using the following data:

30, 32, 11, 14, 40, 37, 16, 26, 12, 33, 13, 19, 38, 12, 28, 15, 39, 11, 37, 17, 27, 14, 36

Interval 11 - 15:

Interval 16 - 20:

Interval 21 - 25:

Interval 26 - 30:

Interval 31 - 35:

Interval 36 - 40:

Montarello and Martins (2005) found that fifth grade students completed more mathematics problems correctly when simple problems were mixed in with their regular math assignments. To further explore this phenomenon, suppose that a researcher selects a standardized mathematics achievement test that produces a normal distribution of scores with a mean of µ= 100 and a standard deviation of σ = 18. The researcher modifies the test by inserting a set of very easy problems among the standardized questions and gives the modified test to a sample of n = 36 students. If the average test score for the sample is M = 104, is this result sufficient to conclude that inserting the easy questions improves student performance? Use a one-tailed test with α = .01.

A)The alternative hypotheses in words is

B)The null hypothesis in symbols is

C)The critical z values is

D)The z-score statistic is:

E) Your decision is

A 1996 study identified 102 out of 852 men as left-handed vs 76 out of 839 women as left handed. The same study also looked at the difference in intelligence between right- and left-handed people. In particular, the average IQ of all right-handed persons in the sample (men and women) was x1 = 97.7, with a standard deviation of s1 = 15.4, whereas the average IQ of all left-handed persons in the sample (men and women) was x2 = 100.3, with a standard deviation of s1 = 17.6. (a) Set up a two-sided hypothesis test on a difference of means and determine the p-value for this sample data. (b) Can it be concluded that left-handed individuals are smarter than right-handed individuals given a significance level of α = 0.05? (c) What is the smallest significance level α (rounded to the nearest hundredth) for which we may conclude that lefties are smarter than righties?

Please show work - trying to understand how to do this so I can apply it to other problems.

Both Professor X and Professor Y agree that students who study for their examinations do better than those who do not. Both professors teach the same class, take a careful four level ranking of how hard their students studied and give a very similar examination.

This year the average score in Professor X’s class and the average score in Professor Y’s class by their level of study effort was virtually THE SAME:

Students who studied seriously scored an average of 100 points

Students who studied moderately earned an average of 80 points

Students who studies briefly earned an average of 60 points

Students who did not study earned an average of 40 points

Professor X                                                                               Professor Y

Class size   100                                                                         class size 100

20% of students Studied Seriously                                             40% of students studied seriously

40% of students studied moderately                                           40% of students studied moderately

30% of students studied briefly                                                  10% of students studies briefly

10% of students did not study at all                                            10% of students did not study at all

What is the average score in Professor X’s class? In Professor Y’s class?

What would be the average scores if both Professors had the SAME composition of students (in relation to the level of effort in studying)

Draw a parse tree for the string “big Jim ate green cheese” in ? = (?, ∑, ?, ?, ), where: ? = {?, ?, ?, ?, ?} ∪ ∑ , ∑ = {???, ???, ?????, ?ℎ????, ???}, ? = {? → ?, ? → ??, ? → ???, ? → ???, ? → ?????, ? → ?ℎ????, ? → ???, ? → ???}.