4. Suppose that it takes an engineer T hours to repair a router, where T ∼ Exp(1/2). (a) Find the probability that it takes the engineer more than 2 hours to repair the router. (b) On May 1st, your router was broken at 5 pm and the engineer starts repairing at 6 pm. While he is repairing the router, you are working on your ECI 114 homework 3. However, the router has not been fixed when you finish the homework, which is at 10 pm. You do know that the homework is due 11:59 pm on Canvas. What is the probability that you do miss the submission? So, do submit the your works early!

(1 point) A researcher is interested in whether the number of years of formal education is related to a person's decision to never smoke, continue to smoke, or quit smoking cigarettes. The data below represent the smoking status by level of education for residents of the United States 18 years or older from a random sample of 350 residents. Round all numeric answers to four decimal places.

1. Select the name of the test that should be used to assess the hypotheses:

?0H0: "Smoking Status" is independent of "Education Level"

??HA: "Smoking Status" is not independent of "Education Level"

A. ?2X2 test of independence
B. ?2X2 test of a single variance
C. ?2X2 goodness of fit

2. Under the null hypothesis, what is the expected number for people with an education of Some college and a smoking status of Never?  

3. Calculate the ?2X2 test statistic.

4.What was the contribution of Never smokers who attended Some college toward this test statistic?

5. What are the degrees of freedom for this test?

6. What is the p-value for this test?

7. Based on the p-value, we have:
A. strong evidence
B. very strong evidence
C. some evidence
D. little evidence
E. extremely strong evidence
that the null model is not a good fit for our observed data.

8. Which of the following is a necessary condition in order for the hypothesis test results to be valid? Check all that apply.

A. There must be at least 10 "yes" and 10 "no" observations for each variable.
B. The observations must be independent of one another.
C. The population data must be normally distributed.
D. There must be an observed count of at least 5 in every cell of the table.
E. There must be an expected count of at least 5 in every cell of the table.

Engineers must consider the breadths of male heads when designing helmets. The company researchers have determined that the population of potential clientele have head breadths that are normally distributed with a mean of 7.1-in and a standard deviation of 1.1-in.

In what range would you expect to find the middle 68% of most head breadths?
Between_______ and ________

If you were to draw samples of size 48 from this population, in what range would you expect to find the middle 68% of most averages for the breadths of male heads in the sample?
Between________ and ________

Enter your answers as numbers. Your answers should be accurate to 2 decimal places.

A survey of company executives reveals that over half of the executives feel that more work is accomplished within their companies on Tuesday than on any other day of the week. To test this claim within his own company, a health insurance company executive takes a random sample of eight employees and records the total number of claims processed by each employee over four Mondays and four Tuesdays. The following gives the recorded data:

Using the five-step format, test whether the claim that more work is accomplished on Tuesdays is correct for this company. Make sure to clearly state the null and alternative hypotheses, define the test statistic and null distribution, calculate the test statistic, find bounds the p-value, and make a conclusion in terms of the problem. (Hint: This data is paired.)

"5 Step Format"

1. State H0 and Ha.

2. State α.

3. State the form of the 1 − α confidence interval you will use, along with all the assumptions necessary.

4. Calculate the 1 − α confidence interval.

5. Based on the 1 − α confidence interval, either: ❑ Reject H0 and conclude Ha, or ❑ Fail to reject H0.

R Programming Exercise 3.4

From a normal distribution which has a standard deviation of 40 and mean of 10, generate 2 to 600 samples. After generating the samples utilize the plot command to plot the mean of the generated sample (x-axis) against the number of samples (Y-axis). Use proper axis labels. Create a second plot of the density of the 600 samples that you generated.

Use adequate comments to explain your reasoning.
This code can be solved in 4 to 8 lines.

For this problem use the following variables:

For the mean use: Mean_of_data

For Standard Deviation use: Standard_deviation_of_data

Please do as little intermediate rounding as possible in order to get the correct p-value.

In a sample of 200 of Deadpool's severed arms, 110 of them grow back a new Deadpool. In a sample of 210 of Deadpool's legs, 90 of them grow back a new Deadpool.  

(a) Do we have evidence at the various levels that Deadpool's severed arms are more likely to grow back a new Deadpool than his severed legs?
The associated p-value for this hypothesis test is ( )? (Answers to four places after the decimal.)

(b)Do we have evidence at the various levels that Deadpool's severed arms and his severed legs have a different probability of growing back a new Deadpool?
The associated p-value for this hypothesis test is ( )?

2.

Clark Heter is an industrial engineer at Lyons Products. He would like to determine whether there are more units produced on the night shift than on the day shift. The mean number of units produced by a sample of 53 day-shift workers was 355. The mean number of units produced by a sample of 62 night-shift workers was 363. Assume the population standard deviation of the number of units produced on the day shift is 24 and 32 on the night shift.

At the 0.01 significance level, is the number of units produced on the night shift larger?

1. Is this a one-tailed or a two-tailed test?

• One-tailed test or Two-tailed test

2. State the decision rule. (Negative value should be indicated by a minus sign. Round your answer to 2 decimal places.)

3. Compute the value of the test statistic. (Negative value should be indicated by a minus sign. Round your answer to 2 decimal places.)

4. What is your decision regarding H₀?

• Reject H₀. or Do not reject H₀.

A clinical trial tests a method designed to increase the probability of conceiving a girl. In the study 532 babies were​ born, and 266 of them were girls. Use the sample data to construct a 99​% confidence interval estimate of the percentage of girls born.

__________<p<______________

​(Round to three decimal places as​ needed.)

What is the optimal time for a scuba diver to be on the bottom of the ocean? That depends on the depth of the dive. The U.S. Navy has done a lot of research on this topic. The Navy defines the "optimal time" to be the time at each depth for the best balance between length of work period and decompression time after surfacing. Let x = depth of dive in meters, and let y = optimal time in hours. A random sample of divers gave the following data.

(a) Find Σx, Σy, Σx², Σy², Σxy, and r. (Round r to three decimal places.)

(c) Find a, and b. (Round your answers to four decimal places.)

(d) Find the predicted optimal time in hours for a dive depth of x = 30 meters. (Round your answer to two decimal places.)
hr

(f) Use a 1% level of significance to test the claim that β < 0. (Round your answers to two decimal places.) Hint 1: The standard error of b is 0.007058. Hint 2: Your answers to the t and critical t should have the same sign.

Conclusion

Fail to reject the null hypothesis. There is insufficient evidence that β < 0.

Reject the null hypothesis. There is insufficient evidence that β < 0.

Fail to reject the null hypothesis. There is sufficient evidence that β < 0.

Reject the null hypothesis. There is sufficient evidence that β < 0.

Let x be the age of a licensed driver in years. Let y be the percentage of all fatal accidents (for a given age) due to failure to yield the right of way. For example, the first data pair states that 5% of all fatal accidents of 37-year-olds are due to failure to yield the right of way.

Complete parts (a) through (e), given Σx = 372, Σy = 112, Σx² = 24814, Σy² = 3118, Σxy = 8224, and r ≈ 0.955.

(b) Verify the given sums Σx, Σy, Σx², Σy², Σxy, and the value of the sample correlation coefficient r. (Round your value for r to three decimal places.)

(c) Find x, and y. Then find the equation of the least-squares line  = a + bx. (Round your answers for x and y to two decimal places. Round your answers for a and b to three decimal places.)

(d) Graph the least-squares line. Be sure to plot the point (x, y) as a point on the line.

(e) Find the value of the coefficient of determination r². What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained? (Round your answer for r² to three decimal places. Round your answers for the percentages to one decimal place.)

(f) Predict the percentage of all fatal accidents due to failing to yield the right of way for 70-year-olds. (Round your answer to two decimal places.)
%

A leisure company operates three amusement arcades in the UK: at Redcar, Skegness and Torquay. As part of a performance review the duration in minutes of the period spent in the arcades by each of a sample of customers visiting was recorded.

The durations of visits made by 21 customers visitng the Redcar arcade were :

23 8 39 72 73 13 44 74 37 37 21

21 27 27 34 31 32 43 74 44 36 36 23  

The figures for 18 customers visiting the Skegness arcade were:

31 51 69 12 53 28 36 28 36 30

35 45 48 25 9 32 60 66

The figures for 20 customers visiting the Torquay arcade were:

3 19 1 15 21 9 7 20 10 2

6 2 11 37 10 6 10 14 3 5

(A) Classify both sets of data into grouped frequency distributions

(B) Calculate the relative frequency for each class of all three distributions

(C) The company expects customers to spend at least 20 minutes on visits to their arcades.

Use your relative frequency figures to compare the performances of the arcades in this respect.

51% of students entering four-year colleges receive a degree within six years. Is this percent smaller than for students who play intramural sports? 123 of the 292 students who played intramural sports received a degree within six years. What can be concluded at the level of significance of αα = 0.01?

1. For this study, we should use Select an answer t-test for a population mean z-test for a population proportion
2. The null and alternative hypotheses would be:
   Ho: ? p μ  Select an answer > < ≥ ≠ ≤ =   (please enter a decimal)   
   H₁: ? p μ  Select an answer = ≥ > < ≠ ≤   (Please enter a decimal)

3. The test statistic ? z t  =  (please show your answer to 3 decimal places.)
4. The p-value =  (Please show your answer to 4 decimal places.)
5. The p-value is ? ≤ >  αα
6. Based on this, we should Select an answer accept fail to reject reject  the null hypothesis.
7. Thus, the final conclusion is that ...
   • The data suggest the population proportion is not significantly smaller than 51% at αα = 0.01, so there is sufficient evidence to conclude that the population proportion of students who played intramural sports who received a degree within six years is equal to 51%.
   • The data suggest the populaton proportion is significantly smaller than 51% at αα = 0.01, so there is sufficient evidence to conclude that the population proportion of students who played intramural sports who received a degree within six years is smaller than 51%
   • The data suggest the population proportion is not significantly smaller than 51% at αα = 0.01, so there is not sufficient evidence to conclude that the population proportion of students who played intramural sports who received a degree within six years is smaller than 51%.
8. Interpret the p-value in the context of the study.
   • There is a 51% chance of a Type I error.
   • There is a 0.12% chance that fewer than 51% of all students who played intramural sports graduate within six years.
   • If the sample proportion of students who played intramural sports who received a degree within six years is 42% and if another 292 such students are surveyed then there would be a 0.12% chance of concluding that fewer than 51% of all students who played intramural sports received a degree within six years.
   • If the population proportion of students who played intramural sports who received a degree within six years is 51% and if another 292 students who played intramural sports are surveyed then there would be a 0.12% chance fewer than 42% of the 292 students surveyed received a degree within six years.
9. Interpret the level of significance in the context of the study.
   • There is a 1% chance that the proportion of all students who played intramural sports who received a degree within six years is smaller than 51%.
   • If the population proportion of students who played intramural sports who received a degree within six years is 51% and if another 292 students who played intramural sports are surveyed then there would be a 1% chance that we would end up falsely concluding that the proportion of all students who played intramural sports who received a degree within six years is smaller than 51%
   • There is a 1% chance that aliens have secretly taken over the earth and have cleverly disguised themselves as the presidents of each of the countries on earth.
   • If the population proportion of students who played intramural sports who received a degree within six years is smaller than 51% and if another 292 students who played intramural sports are surveyed then there would be a 1% chance that we would end up falsely concluding that the proportion of all students who played intramural sports who received a degree within six years is equal to 51%.

You are interested in the development of numeracy in childhood and want to understand the impact a child attending preschool has on their understanding of numbers. You believe that preschool can strongly improve a child’s numeracy skills. You collect data from 8 children using a numeracy measure where an increased score indicates increased numerical ability. You collect data from each child before they start preschool as well as after they complete preschool.

A. Write out your null and alternative hypotheses.

B. Conduct the statistical test using alpha .05

C. Determine whether the result is significant or not and make your decision regarding the null hypothesis.

D. Explain your finding in terms of your research question. In other words, what has this shown us about preschool and numeracy?

Student      Before preschool     After preschool

1                             45                        43

2                             33                        39

3                             46                        50

4                             49                        49

5                             28                        31

6                             43                        46

7                             36                        34

8                             37                        38