Benford's Law states that the first nonzero digits of numbers drawn at random from a large complex data file have the following probability distribution.†

Suppose that n = 275 numerical entries were drawn at random from a large accounting file of a major corporation. The first nonzero digits were recorded for the sample.

Use a 1% level of significance to test the claim that the distribution of first nonzero digits in this accounting file follows Benford's Law.

A. What is the level of significance?

B. State the null and alternate hypotheses.

C. 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.)

D. Are all the expected frequencies greater than 5?

E. What sampling distribution will you use?

F. What are the degrees of freedom?

G. Estimate the P-value of the sample test statistic.

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

You are the foreman of the Bar-S cattle ranch in Colorado. A neighboring ranch has calves for sale, and you are going to buy some calves to add to the Bar-S herd. How much should a healthy calf weigh? Let x be the age of the calf (in weeks), and let y be the weight of the calf (in kilograms). x 1 5 11 16 26 36 y 39 47 73 100 150 200 Complete parts (a) through (e), given Σx = 95, Σy = 609, Σx2 = 2375, Σy2 = 81,559, Σx y = 13,777, and r ≈ 0.997. (a) Make a scatter diagram of the data.

(Select the correct graph.) A scatter diagram with 6 points is graphed on the x y coordinate plane. The points are located at (1, 39), (5, 47), (11, 73), (16, 100), (26, 150), (36, 200). When considering the data points as a whole, the points are loosely gathered into a mass that is lower on the left and higher on the right. A scatter diagram with 6 points is graphed on the x y coordinate plane. The points are located at (1, 29), (5, 37), (11, 63), (16, 90), (26, 140), (36, 190). When considering the data points as a whole, the points are loosely gathered into a mass that is lower on the left and higher on the right. A scatter diagram with 6 points is graphed on the x y coordinate plane. The points are located at (3, 39), (7, 47), (13, 73), (18, 100), (28, 150), (38, 200). When considering the data points as a whole, the points are loosely gathered into a mass that is lower on the left and higher on the right. A scatter diagram with 6 points is graphed on the x y coordinate plane. The points are located at (3, 29), (7, 37), (13, 63), (18, 90), (28, 140), (38, 190). When considering the data points as a whole, the points are loosely gathered into a mass that is lower on the left and higher on the right. (b) Verify the given sums Σx, Σy, Σx2, Σy2, Σx y, and the value of the sample correlation coefficient r. (For each answer, enter a number. Round your value for r to three decimal places.) Σx = Σy = Σx2 = Σy2 = Σx y = r = (c) Find x bar, and y bar. Then find the equation of the least-squares line y hat = a + b x. (For each answer, enter a number. Round your answers for x bar and y bar to two decimal places. Round your answers for a and b to three decimal places.) x bar = x bar = y bar = y bar = y hat = value of a coefficient + value of b coefficient x (d) Graph the least-squares line. Be sure to plot the point (x bar, y bar) as a point on the line. (Select the correct graph.) A line and a point are graphed on the x y coordinate plane. The point is located the approximate point (15.8, 102). The line enters the window at approximately y = 171 on the positive y axis, goes down and right, passes through the approximate point (15.8, 102), and exits the window at approximately x = 39.1 on the positive x axis. A line and a point are graphed on the x y coordinate plane. The point is located the approximate point (15.8, 132). The line enters the window at approximately y = 201 on the positive y axis, goes down and right, passes through the approximate point (15.8, 132), and exits the window in the first quadrant. A line and a point are graphed on the x y coordinate plane. The point is located the approximate point (15.8, 132). The line enters the window at approximately y = 56 on the positive y axis, goes up and right, passes through the approximate point (15.8, 132), and exits the window in the first quadrant. A line and a point are graphed on the x y coordinate plane. The point is located the approximate point (15.8, 102). The line enters the window at approximately y = 26 on the positive y axis, goes up and right, passes through the approximate point (15.8, 102), and exits the window in the first quadrant. (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? (For each answer, enter a number. Round your answer for r2 to three decimal places. Round your answers for the percentages to one decimal place.) r2 = explained = % unexplained = % (f) The calves you want to buy are 21 weeks old. What does the least-squares line predict for a healthy weight (in kg)? (Enter a number. Round your answer to two decimal places.) kg

Accrotime is a manufacturer of quartz crystal watches. Accrotime researchers have shown that the watches have an average life of 26 months before certain electronic components deteriorate, causing the watch to become unreliable. The standard deviation of watch lifetimes is 4 months, and the distribution of lifetimes is normal.

(a) If Accrotime guarantees a full refund on any defective watch for 2 years after purchase, what percentage of total production will the company expect to replace? (Round your answer to two decimal places.)


(b) If Accrotime does not want to make refunds on more than 12% of the watches it makes, how long should the guarantee period be (to the nearest month)?

The concerns about the impact of technology on our lives are growing. There have been recent research reports about the negative impact of digital technology usage on​ stress, work​ productivity, happiness, and overall​ well-being. In light of these​ concerns, a polling organization asked technology experts about their opinion on our digital life and found that 29% of the experts believe that in the future our lives will be more harmed than helped by their digital environment. In an effort to confirm these​ results, a local tech company decides to ask 190 randomly selected customers their opinions about their digital life. Answer parts a through c below.

A. What is the probability that 35 or more people from this sample are concerned about the digital​ environment?

The probability is =

​(Round to four decimal places)

B. What is the probability that 45 or more people from this sample are concerned about the digital​ environment?

The probability is =

​(Round to four decimal places)

(please round to the correct decimal places)

Give one strength and one limitation of each these study designs for studying a disease and an exposure: randomized clinical trial, population-based (cross-sectional) study and case-control study (disease-based). Which study design is better for answering a research about the possible casual effects of an exposure on disease based on comparing the strengths and limitations you listed?

Study Design: Randomized control trial

Strength(s):

Limitation(s):

Study Design: Cross-sectional

Strength(s):

Limitation(s):

Study Design: Disease-based

Strength(s):

Limitation(s):

Study design that is best for evaluating possible casual effects of an exposure on disease:

Justify choice:

A sample of 114 patients were given a drug to lower cholesterol. A 95% confidence interval for the mean reduction in cholesterol (in mmol/L) was (0.88, 1.02). What was the sample standard deviation of the reduction amounts? The standard deviation was ? mmol/L.

a. The following are the GPA's of a class of Stats students: 3.4, 3.1, 2.0, 2.3, 3.3, 2.6, 1.8, 0.2, 3.1, 4.0, 0.7, 2.3, Construct the 99% CI.

b.Would a 90% CI be larger or smaller?

Please submit your Excel file and highlight your answers in color. Brief Case Has Gold Lost its Luster? In 2011, when the Gallup organization polled investors, 34% rated gold the best long-term investment. However, in April of 2013 Gallup surveyed a random sample of U.S. adults. Respondents were asked to select the best long-term investment from a list of possibilities. Only 241 of the 1005 respondents chose gold as the best long-term investment.

A. With 95% confidence, compute the margin of error of the sample proportion.

B. Compute and describe a 95% confidence interval in the context of the case.

C. Do you think opinions about the value of gold as a long-term investment have really changed from the old 34% favorable rate, or do you think this is a sample variability? Explain your answer using the calculated statistics.

D. Suppose the Gallup organization wants to offer a new investment option and wants to estimate, to within 5%, the proportion of customers who are likely to make this new investment with 95% confidence. How large a sample do they need?

Drug A is the usual treatment for depression in graduate students. Pfizer has a new drug, Drug B, that it thinks may be more effective. You have been hired to design the test program. As part of your project briefing, you decide to explain the logic of statistical testing to the people who are going to be working for you.

Write the research hypothesis and the null hypothesis.

Then construct a table like the one below, displaying the outcomes that would constitute Type I and Type II error.

Write a paragraph explaining which error would be more severe, and why.

An engineer has designed a valve that will regulate water pressure on an automobile engine. The valve was tested on 140 engines and the mean pressure was 7.7 lbs/square inch. Assume the variance is known to be 0.64. If the valve was designed to produce a mean pressure of 7.9lbs/square inch, is there sufficient evidence at the 0.1 level that the valve does not perform to the specifications?

State the null and alternative hypotheses for the above scenario.

Researchers measured the amount of plasma carotenoids (in micrograms per milliliter) of male and female zebra finches. Here are some summary statistics from the study (note: standard error=standard deviation/sqrt(n)):

Do male and female zebra finches differ in their plasma carotenoid levels? To answer this question, the researchers tested the following hypotheses:

H₀: m_M = m_F versus H_a: m_M ≠ m_F

What can we conclude at significance level 5%?

1. Reject the alternative hypothesis
2. Fail to reject the alternative hypothesis
3. Reject the null hypothesis
4. Fail to reject the null hypothesis

Researchers aim to study the weights of 10-year-old girls living in the United States (possibly to compare to other countries and thus compare growth rates). Based on previous studies, we can assume that weights of 10-year-old girls are Normal. From a small sample of 16 girls, the researchers find a sample average of ?̅ = 91.4 pounds and a sample standard deviation of ? = 2.8 pounds. Create a 99% confidence interval for the true average weight of 10-year-old girls in the U.S. (and make a formal conclusion based on your calculated interval).

A newsgroup is interested in constructing a 90% confidence interval for the proportion of all Americans who are in favor of a new Green initiative. Of the 588 randomly selected Americans surveyed, 354 were in favor of the initiative. Round answers to 4 decimal places where possible.

a. With 90% confidence the proportion of all Americans who favor the new Green initiative is between ___?__ and __?___

b. If many groups of 588 randomly selected Americans were surveyed, then a different confidence interval would be produced from each group. About ___?___ percent of these confidence intervals will contain the true population proportion of Americans who favor the Green initiative and about ___?___ percent will not contain the true population proportion.

A journalist once wrote about Tiger Woods’ ability to detect subtle differences in golf equipment. Tiger Woods was sent four golf clubs to test. The four clubs looked identical, but one was heavier than the rest by just two grams (about the weight of a dollar bill). Tiger Woods swung each of the four clubs and quickly declared, “This one’s heavier.” He was right.

Suppose this basic test is carried out for a random sample of 75 professional golfers. Each golfer swings the four clubs and has to decide which club is heavier, and 23 out of the 75 golfers pick out the correct club as being heavier.   We want to conduct a hypothesis test where our null hypothesis is that these golfers are just guessing and the alternative hypothesis is that a greater proportion (p) of professional golfers than expected under random chance can recognize the heaviest of the four clubs. In symbols, our hypotheses would be

H_o: p = 0.25

H_a: p > 0.25

1. Why is the null hypothesis that p = 0.25?

2. From the above information, what will the sample proportion (or p) be? Please compute this below and round to two decimal places.

3. Use the following formula to compute the test statistic.

z=p-pp(1-p)n

4. Based on what you see upon looking up the test statistic in Table B, what should the p-value be?

5. If our significance or alpha level is 0.05, should we consider the results of this hypothesis test to be statistically significant? Please explain why or why not.

Use the t-distribution to find a confidence interval for a mean μ given the relevant sample results. Give the best point estimate for μ, the margin of error, and the confidence interval. Assume the results come from a random sample from a population that is approximately normally distributed.
A 99% confidence interval for μ using the sample results x¯=82.7, s=32.9, and n=15
Round your answer for the point estimate to one decimal place, and your answers for the margin of error and the confidence interval to two decimal places.

Point estimate=

Margin of error=

The 99% confidence interval is _____ to ______.