A researcher believes that the proportion of male high school seniors who have their own cars is higher than the proportion of female high school seniors who have their own cars. In a sample of 35 male high school seniors 20 had their own cars. In a sample of 40 female high school seniors, 20 had their own cars. At α = 0.05, test the researchers claim using the five step testing method.

The state of Virginia shows than automobile is driven an average 23,500 km/year with a stand deviation of 3900 kilometers. Assume the distribution of measurements is approximately normal.

a) What is the probability that a randomly selected automobile is found to have driven at most 25,000km/yr?

b) What is the probability that a randomly selected automobile is found to have driven between 20,000 and 30,000 km/yr?)

c) Solve for the third quartile.

d) Suppose that the average cost of leasing a car is $300 per month with a standard deviation of $110. You are interested in leasing a car and decide to investigate the market. You randomly sample 30 people among your acquaintances that are currently leasing. What is the probability that among those 30 people the sample mean is less than $260?

e) What is the probability that among the 30 people sampled in part d) the sample mean is greater than $330?

Historically, evening long-distance calls phone calls from a particular city have averages 12 minutes per call. In a random sample of 20 calls, the sample mean was 10.7 minutes per call with a standard deviation of 4 minutes. Does the sample indicate a change in the mean duration of long distance calls? Test the hypothesis at 10% level of significance, state the hypothesis and report your conclusion.

a. estimate the p-value and show your work.

[02] For n ≥ 1, how many strings of length n using letters a,b,c are there if the letter a must occur an even number of times?

The observations are listed : 5.32, 9.87, 11.25, 10.94, 5.58, 6.29, 7.47, 10.75, 6.22, 8.00. Apply backward empirical rule, IQR/S, and normal probability plot to check the normality assumption.

A professional astrologer has prepared horoscopes of 83 adults. Each
adult was shown three horoscopes, one of which was the one the astrologer prepared for them
while the other two were randomly chosen from those belonging to the other subjects. Each
adult had to guess which of the three horoscopes was their's. The astrologer claims that the
probability of correct prediction (say, p) is higher than that corresponding to mere guessing.
Of the 83 subjects, 28 guessed their horoscope correctly.

Suppose you construct a 99% confidence interval of p, the probability of
guessing the right horoscope. This interval will contain the null value of p0 = 1/3.

A. True
B. False

The following table shows ceremonial ranking and type of pottery sherd for a random sample of 434 sherds at an archaeological location.

Use a chi-square test to determine if ceremonial ranking and pottery type are independent at the 0.05 level of significance.

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: Ceremonial ranking and pottery type are not independent.
H₁: Ceremonial ranking and pottery type are not independent.H₀: Ceremonial ranking and pottery type are not independent.
H₁: Ceremonial ranking and pottery type are independent.    H₀: Ceremonial ranking and pottery type are independent.
H₁: Ceremonial ranking and pottery type are not independent.H₀: Ceremonial ranking and pottery type are independent.
H₁: Ceremonial ranking and pottery type are independent.

(b) Find the value of the chi-square statistic for the sample. (Round the expected frequencies to at least three decimal places. Round the test statistic to three decimal places.)


Are all the expected frequencies greater than 5?

YesNo    

What sampling distribution will you use?

uniformchi-square    normalbinomialStudent's t

What are the degrees of freedom?


(c) Find or estimate the P-value of the sample test statistic. (Round your answer to three decimal places.)

p-value > 0.1000.050 < p-value < 0.100    0.025 < p-value < 0.0500.010 < p-value < 0.0250.005 < p-value < 0.010p-value < 0.005

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

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 sufficient evidence to conclude that ceremonial ranking and pottery type are not independent.At the 5% level of significance, there is insufficient evidence to conclude that ceremonial ranking and pottery type are not independent.    

1. A case-control study was performed to determine whether head injury was associated with an increased risk of brain tumors in children. 200 cases with brain cancer were identified from the state cancer registry and 200 controls were recruited from the same neighborhoods where the cases lived. The mothers of the children completed a questionnaire that asked them to describe their child's past history of head injury. The investigators found that the mothers of the children with brain tumors reported a past head injury for 70 of the cases while a past history of head injury was reported in 30 of the controls. What type of bias was likely to have influenced the findings of this study?

2-After investigators calculate a measure of association (RR for cohort studies and OR for case-control studies), they must evaluate whether or not the observed result is true.

3-Bias in epidemiologic studies means that the investigator is "prejudiced".

4-Control selection bias occurs in a case-control study when cases remember or report their exposures differently (more or less accurately) from controls.

NYU is testing out two different versions of filtering software in order to reduce spam emails. The old version is called "Spam-A-Lot" and the new version is called "Spam-A-Little." In testing each version of the software the following data was produced:

Let p₁ and p₂ denote the true proportion of unsolicited mail that make it through the "Spam-A-Lot" and "Spam-A-Little" filters, respectively.

(a) Determine the unbiased point estimates of p₁ and p₂:

(b) Explain why the formula for a large-sample confidence interval estimate for p1 - p2 can be used in this case.

(c) Build a 95% confidence interval for the true decrease in proportion p1 - p2 of unsolicited mail by switching filters from "Spam-A-Lot" (p₁) to "Spam-A-Little" (p₂), using the sample values given. Record results to 4 decimals.

(d) Based on your answer to (c), has the new filtering program reduced the amount of spam?

(e) Complete the following to perform a hypothesis test at the 5% significance level to test the claim that switching to the new filter "Spam-A-Little" has decreased the proportion of unsolicited emails getting through the filter.

i) H₀:

H_a:

Level of Significance:

Observed Test Statistic (z-statistic):

ii) p-value:

Decision with justification:

Conclusion in context:

210 students were asked to randomly pick one of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. The number 7 was picked by 61 students.

(c) Calculate a 90% confidence interval for the population proportion. (Round the answer to three decimal places.)

___________to ___________

(d) Calculate a 95% confidence interval for the population proportion. (Round the answer to three decimal places.)
to  

__________to____________

(e) Calculate a 98% confidence interval for the population proportion. (Round the answer to three decimal places.)

____________to_________

When σ is unknown and the sample is of size n ≥ 30, there are two methods for computing confidence intervals for μ.

Method 1: Use the Student's t distribution with d.f. = n − 1.
This is the method used in the text. It is widely employed in statistical studies. Also, most statistical software packages use this method.

Method 2: When n ≥ 30, use the sample standard deviation s as an estimate for σ, and then use the standard normal distribution.
This method is based on the fact that for large samples, s is a fairly good approximation for σ. Also, for large n, the critical values for the Student's t distribution approach those of the standard normal distribution.

Consider a random sample of size n = 41, with sample mean x = 45.4 and sample standard deviation s = 6.0.

(a) Compute a 99% confidence intervals for μ using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

(b) Compute a 99% confidence interval for μ using Method 2 with the standard normal distribution. Use s as an estimate for σ. Round endpoints to two digits after the decimal.

(d) Now consider a sample size of 71. Compute a 99% confidence interval for μ using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

(e) Compute a 99% confidence interval for μ using Method 2 with the standard normal distribution. Use s as an estimate for σ. Round endpoints to two digits after the decimal.

(f) Compare intervals for the two methods. Would you say that confidence intervals using a Student's t distribution are more conservative in the sense that they tend to be longer than intervals based on the standard normal distribution?

An animal can be in one of two states A or B. An individual in state A will change to state B at an exponential rate α=1； an individual in stat B divides into two new individuals of type A at rate β=2.

At the beginning time 0, there is one cell, and it is type A. Estimate the expected number of cells of each type at time 5 and time 10.

Cell phone talk time between charges is advertised as 10 hours. Assume that talk time is approximately normally distributed with a mean of 10 hours and a standard deviation of 0.75 hours. a. Find the probability that talk time between charges for a randomly selected cell phone is above 11.35 hours.

2.b. Your phone gets only about 8.35 hours of talk time, what proportion of phones gets less talk time than yours?

2.c. How much talk time would you get between charges, if you had a phone that get just into the top 1%?

Does changing the cutoff value for a prediction model change the Cumulative Life Chart? Explain your answer.

Imagine that we have a population that is positively (right) skewed that has a mean of 114 and a standard deviation of 12. Using a computer simulation program, Jake creates a sampling distribution of size n=5 from this population and Alex creates a sampling distribution of size n=40 from this population. How would Alex’ and Jake’s sampling distributions compare to each other?

Group of answer choices

They would have the same mean and shape, but would differ in standard error.

They would have the same shape, but would differ in mean and standard error.

They would be exactly the same since they are drawn from the same population.

They would have the same mean, but would differ in shape and standard error.