Consider the following hypothesis test:

H₀: p ≥ 0.75
H_a: p < 0.75

A sample of 300 items was selected. Compute the p-value and state your conclusion for each of the following sample results. Use  = .05.

Round your answers to four decimal places.

b. = 0.74

p-value is

c. = 0.78

p-value is

A survey of high school girls classified them by two attributes: whether or not they participated in sports and whether or not they had one or more older brothers.

1. Use the following data to test the null hypothesis that these two attributes of classification are independent.
2. Suppose we do the below hypothesis test instead:

where,

p1-Propotion of young girls participated in sports of those who has a brother

p2-Propotion of young girls participated in sports of those who does not have a brother.

What would be the test statistic and the p value for this test?

A random survey of 100 students asked each student to select the most preferred form of recreational activity from five choices. Following are the results of the survey:

1. Test if the choice of spot and gender are independent (You may use 5% significance level).
2. Calculate the contribution from men who prefer basketball towards the final chi-square test statistic.
3. Re-do the test using MINITAB. Compare with the test statistic and the p value from hand calculations.

Let ξ and η be independent of each other, ξ follows the poisson distribution, and η follows N (0,1).

Prove that ξ + η is a continuous random variable

To predict a bear’s weight, data was taken from 54 anesthetized bears on their age, gender (M=1, F=2), head width_, neck size,  overall length, and chest size.

Use above data to construct a 95% confidence interval for the variable ' AGE' and answer following:

Name the critical value needed to compute the error

Compute the Margin of Error (3 decimal places)  

Name the lower limit of the confidence interval (3 decimal places)

Name the upper limit of the confidence interval (3 decimal places)

What percent of variation(1 decimal place) in a bear's weight is NOT explained by the combination of variables in the regression equation?

In a study, people were observed for about 10 seconds in public places, such as malls and restaurants, to

          determine whether they smiled during the randomly chosen 10-second interval.  The researchers observed

          600 people who were aged 20 years or younger, and 47% of these people smiled.  The researchers also

          observed 975 people who were aged 21 years or older, and 32% of these people smiled.  (Treat this data

          as if it were two simple random samples.)

          a)  Fill in the following table.

          b)  Determine and interpret the 95% confidence interval estimate of the difference between the population

              proportions.

          c)  What does this suggest about how often people in these age groups smile?  Does one group smile

               significantly more often than the other?  If so, which one?

We interviewed two groups of 50 college students respectively from UIC and DePaul to know if they rather watch NFL Football vs. some other sport on Sunday.

Actual Data

Based on the statistics above, we need to determine if there is a relationship between the university of a student and watching Football. Answer to the following questions to arrive at the conclusion.

• What are Ho and Ha?
• What are the chi-square values?
• What is the degrees of freedom?
• What is the p-value?
• For alpha=0.05, what would be our statistical conclusion?
• Write out the full conclusion in English

Please show work

1. A machine produces metal rods used in an automobile suspension system. A random sample of 12 rods is selected and the diameter is measured. The resulting data, in mm, are shown here:

3.11     2.88     3.08     3.01

2.84     2.86     3.04     3.09

3.08     2.89     3.12     2.98

a.         Find a two-sided 95% confidence interval for the mean rod diameter. State the assumption necessary to find the confidence interval.  

2. Is there any evidence to indicate that mean rod diameter is different from 2.95 at significance level a = 0.05? Write the appropriate hypothesis. Make required assumption(s).
3. Can you make the same decision as in part b. from the result of part a. without testing the Hypothesis? If yes, how?

Based on interviews with 82 SARS​ patients, researchers found that the mean incubation period was 4.7 ​days, with a standard deviation of 14.6 days. Based on this​ information, construct a​ 95% confidence interval for the mean incubation period of the SARS virus. Interpret the interval.

lower bound ___ days

upper bound ___ days

Suppose a simple random sample of size n=41 is obtained from a population with mu=61 and sigma=19.

(b) Assuming the normal model can be​ used, determine

​P(x overbarx < 71.4).

​(c) Assuming the normal model can be​ used, determine

​P(x overbarx ≥ 69.1).

Using the following data (already sorted), use a goodness of fit test to test whether it comes from an exponential distribution. The exponential distribution has one parameter, its mean, μ (which is also its standard deviation). The exponential distribution is a continuous distribution that takes on only positive values in the interval (0,∞). Probabilities for the exponential distribution can be found based on the following probability expression:

.

Use 10 equally likely cells for your goodness of fit test.

Data Display

0.2 0.4 0.5 0.5 0.7 0.8 1.0 1.2 1.2 1.2

1.4 1.5 1.5 1.6 1.7 1.7 1.7 1.8 1.8 1.9

2.0 2.3 2.6 2.7 2.7 2.8 2.8 2.8 2.8 2.8

2.8 2.9 2.9 3.0 3.0 3.0 3.2 3.2 3.2 3.4

3.4 3.5 3.6 3.6 3.7 3.8 3.9 3.9 3.9 4.0

4.1 4.1 4.2 4.3 4.5 4.5 4.5 4.6 4.7 4.8

4.8 4.9 4.9 4.9 5.0 5.0 5.1 5.1 5.1 5.3

5.3 5.3 5.3 5.4 5.4 5.4 5.4 5.5 5.5 5.5

5.6 5.6 5.6 5.7 5.7 5.8 5.8 5.8 5.9 5.9

6.0 6.0 6.2 6.2 6.2 6.3 6.3 6.3 6.3 6.4

6.6 6.6 6.6 6.6 6.7 6.8 6.9 6.9 6.9 7.0

7.0 7.1 7.2 7.3 7.3 7.4 7.5 7.5 7.6 7.6

7.7 7.8 7.8 7.9 8.0 8.0 8.0 8.1 8.1 8.1

8.2 8.3 8.4 8.4 8.4 8.5 8.5 8.6 8.6 8.7

8.7 8.8 9.0 9.1 9.1 9.2 9.3 9.4 9.5 9.6

9.6 9.6 9.8 9.9 9.9 9.9 10.0 10.1 10.2 10.5

10.6 10.7 10.7 10.8 10.9 10.9 11.0 11.0 11.4 11.5

11.7 11.8 11.8 11.9 12.0 12.0 12.1 12.1 12.3 12.3

12.3 12.3 12.6 12.9 13.1 13.3 13.3 13.4 13.5 13.6

13.9 14.0 14.2 14.2 14.3 14.3 14.4 15.0 15.0 15.2

15.6 15.6 15.7 15.7 15.7 15.9 16.0 16.3 16.4 16.5

16.5 16.6 16.6 16.7 17.2 17.3 17.3 17.4 17.7 17.9

18.6 18.8 19.9 19.9 19.9 20.0 20.1 20.3 20.4 21.0

21.3 21.5 22.2 23.3 23.5 23.9 24.3 24.8 25.5 25.5

25.6 25.8 27.5 28.2 30.9 35.7 36.3 37.2 40.9 52.8

Descriptive Statistics:

Variable    N   Mean

Exp?      250 9.974

SURVIVAL ANALYSIS

QUESTION 1

Indicate whether the following statement is true or false, please explain your answer in one sentence.

(a)        The survival function can be expressed in terms of the cumulative hazard

(b)        The hazard rate in a Weibull model increases with time

(c)        The hazard rate in Pareto model is monotonic function

(d)        In survival analysis, the survival function is always increasing

(e)        In survival analysis if a subject is right censored, then he'she will never experience the event of interest

(f)        The hazard function is a bounded function

The white "Spirit" black bear (or Kermode) Ursus americanus kermodei, differs from the ordinary black bear by a single amino acid change in the melanocortin 1 receptor gene (MC1R).

In this population, the gene has two forms (or alleles): the "white" allele b and the "black" allele B. The trait is recessive: white bears have two copies of the white allele of this gene (bb), whereas a bear is black if it has one or two copies of the black allele (Bb or BB). Both color morphs and all three genotypes are found together in the bear population of the northwest coast of British Columbia. If possessing the white allele has no effect on growth, survival, reproductive success, or mating patterns of individual bears, then the frequency of individuals with BB, Bb, or bb allele combinations in the population will follow a binomial distribution (that is BB- 25%, Bb- 50% and bb- 25%). To investigate, Hedrick and Ritland (2011) sampled and genotyped 87 bears from the northwest coast:

42 were BB

24 were Bb

21 were bb

Assume that this is a random sample.

A formal hypothesis test was carried out to compare the observed and expected frequencies of genotypes.

(a)  (null or alternative) hypothesis would be "The frequency distribution of genotypes has a binomial distribution in the population"

whereas "The frequency distribution of genotypes does not have a binomial distribution" is the  (null or alternative) hypothesis.

(b) The degrees of freedom for the test statistic are .

(c) The calculated chi-square value is  (report to one decimal place)

(d) The critical chi-square value at alpha =0.05 is  (report the whole number from the provided chi-square distribution table)

(e) The difference between the observed and expected frequencies is statistically significant.  (Yes or No)

(f) The calculated chi-square value exceeds the critical chi-square value corresponding to = 0.05.  (Yes or No)

(g) The calculated chi-square value exceeds the critical chi-square value corresponding  to  = 0.01.  (Yes or No)

A person's blood glucose level and diabetes are closely related. Let x be a random variable measured in milligrams of glucose per deciliter (1/10 of a liter) of blood. Suppose that after a 12-hour fast, the random variable x will have a distribution that is approximately normal with mean μ = 90 and standard deviation σ = 30. Note: After 50 years of age, both the mean and standard deviation tend to increase. For an adult (under 50) after a 12-hour fast, find the following probabilities. (Round your answers to four decimal places.)

(a) x is more than 60


(b) x is less than 110


(c) x is between 60 and 110


(d) x is greater than 125 (borderline diabetes starts at 125)

The Wall Street Journal reported that the age at first startup for 25% of entrepreneurs was 29 years of age or less and the age at first startup for 75% of entrepreneurs was 30 years of age or more. (a) Suppose a sample of 200 entrepreneurs will be taken to learn about the most important qualities of entrepreneurs. Show the sampling distribution of p where p is the sample proportion of entrepreneurs whose first startup was at 29 years of age or less. If required, round your answers to four decimal places. np = n(1-p) = E(p) = σ(p) = (b) Suppose a sample of 200 entrepreneurs will be taken to learn about the most important qualities of entrepreneurs. Show the sampling distribution of p where p is now the sample proportion of entrepreneurs whose first startup was at 30 years of age or more. If required, round your answers to four decimal places. np = n(1-p) = E(p) = σ(p) = (c) Are the standard errors of the sampling distributions different in parts (a) and (b)?