Bill Alther is a zoologist who studies Anna's hummingbird (Calypte anna).† Suppose that in a remote part of the Grand Canyon, a random sample of six of these birds was caught, weighed, and released. The weights (in grams) were as follows.

3.7 2.9 3.8 4.2 4.8 3.1

The sample mean is x = 3.75 grams. Let x be a random variable representing weights of hummingbirds in this part of the Grand Canyon. We assume that x has a normal distribution and σ = 0.66 gram. Suppose it is known that for the population of all Anna's hummingbirds, the mean weight is μ = 4.45 grams. Do the data indicate that the mean weight of these birds in this part of the Grand Canyon is less than 4.45 grams? Use α = 0.01.

(a) What is the level of significance?

State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test?

H₀: μ < 4.45 g; H₁: μ = 4.45 g; left-tailed

H₀: μ = 4.45 g; H₁: μ ≠ 4.45 g; two-tailed

H₀: μ = 4.45 g; H₁: μ < 4.45 g; left-tailed

H₀: μ = 4.45 g; H₁: μ > 4.45 g; right-tailed

(b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution.

The Student's t, since we assume that x has a normal distribution with known σ.

The standard normal, since we assume that x has a normal distribution with known σ.   

The standard normal, since we assume that x has a normal distribution with unknown σ.

The Student's t, since n is large with unknown σ.

Compute the z value of the sample test statistic. (Round your answer to two decimal places.)

(c) Find (or estimate) the P-value. (Round your answer to four decimal places.)

Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?
At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.  

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e) State your conclusion in the context of the application.

There is sufficient evidence at the 0.01 level to conclude that humming birds in the Grand Canyon weigh less than 4.45 grams.

There is insufficient evidence at the 0.01 level to conclude that humming birds in the Grand Canyon weigh less than 4.45 grams.    

A random sample of n measurements was selected from a population with unknown mean mu and standard deviation sigma equals 20 for each of the situations in parts a through d. Calculate a 95​% confidence interval for mu for each of these situations.

a. n=60​, overbar =29 ​(Round to two decimal places as​ needed.)

b. n=250​, xoverbare=113 ​(Round to two decimal places as​ needed.)

c. n=90​, xoverbar=18 ​(Round to two decimal places as​ needed.)

d. n=90​, xoverbar=4.1 ​(Round to two decimal places as​ needed.)

When statisticians say that the t Test produces the same thing as the F test, what they really mean is that:

__this occurs when the t Test is used to measure more than 2 groups

__t = F

__t2 = F

__none of the above

Use only the information in the following table for the next few questions.  The following statistics represent samples of copper wire submitted by two companies for tensile strength testing (psi).

Statistic                                                      Company A          Company B

Arithmetic Mean                                       500                         600

Median                                                       500                        500

Mode                                                           500                       300

Standard Deviation                                   40                          20

Mean Absolute Deviation                         32                        16

Quartile Deviation                                     25                         14

Range                                                       240                         120

Sample Size                                          100                           80

The middle 95% of the wires from Company A tested between _____ and _____ .  Record ONLY the lower limit as a whole number.

The middle 50 percent of the wires of Company A tested between _____ and _____.  Record ONLY the lower limit as a whole number.

Which company's distribution has the larger dispersion (answer either 'A' or 'B')?

The variance for Company A is _____?

What is the t distribution and why is it needed?  How do we know to use the t distribution in the construction of a confidence interval of the mean rather than the normal distribution?  Give an example.  Ask a question

An instructor who taught two sections of engineering statistics last term, the first with 25 students and the second with 35, decided to assign a term project. After all projects had been turned in, the instructor randomly ordered them before grading. Consider the first 15 graded projects.

(a) What is the probability that exactly 10 of these are from the second section? (Round your answer to four decimal places.)


(b) What is the probability that at least 10 of these are from the second section? (Round your answer to four decimal places.)


(c) What is the probability that at least 10 of these are from the same section? (Round your answer to four decimal places.)


(d) What are the mean value and standard deviation of the number among these 15 that are from the second section? (Round your mean to the nearest whole number and your standard deviation to three decimal places.)

(e) What are the mean value and standard deviation of the number of projects not among these first 15 that are from the second section? (Round your mean to the nearest whole number and your standard deviation to three decimal places.)

In a statistics class, 8 students took their pulses before and after an exam. The pulse rates (beats per minute) of the students before and after the exam were obtained separately and are shown in the table. Treat this as though it were a random sample of statistics students. Test the hypothesis that the mean of statistics students' pulse rates is higher after an exam using a significance level of 0.05. Do the 5-step hypothesis test and submit an image of your work.

An automobile club reported that the average price of regular gasoline in a certain state was $2.83 per gallon. The following data show the price per gallon of regular gasoline for 15 randomly selected stations in the state. Complete parts a through c below

a. Construct an 80​% confidence interval to estimate the average price per gallon of gasoline in the state.

Answer: ​The 80% confidence interval to estimate the average price per gallon of gasoline in the state is from ​$2.78 per gallon to ​$2.85 per gallon.

b. Do the results from this sample validate the automobile​ club's findings?

Answer: First determine whether the population​ mean, 2.83​, is contained within the 80 % confidence​ interval, which is approximately from 2.78 to 2.85.

Note that 2.83 is contained within the 80 % confidence interval.

To determine whether the results from this sample validate the automobile​ club's findings, recall that a confidence interval for the mean is an interval estimate around a sample mean that provides one with a range of where the true population mean lies.

c. What assumptions need to be made about this​ population?

Answer: Carefully review the requirements for calculating a confidence interval with the​ Student's t-distribution when the sample size is less than or equal to 30.

Above are the questions and answers. Please provide all Excel formulas (not calculations) that get to these answers.

Castaneda v. Partida is an important court case in which statistical methods were used as part of a legal argument. When reviewing this case, the Supreme Court used the phrase "two or three standard deviations" as a criterion for statistical significance. This Supreme Court review has served as the basis for many subsequent applications of statistical methods in legal settings. (The two or three standard deviations referred to by the Court are values of the z statistic and correspond to P-values of approximately 0.05 and 0.0026.) In Castaneda the plaintiffs alleged that the method for selecting juries in a county in Texas was biased against Mexican Americans. For the period of time at issue, there were 180,500 persons eligible for jury duty, of whom 142,600 were Mexican Americans. Of the 893 people selected for jury duty, 347 were Mexican Americans.

(a) What proportion of eligible voters were Mexican Americans? Let this value be p_o. (Round your answer to four decimal places.)
___________

(b) Let p be the probability that a randomly selected juror is a Mexican American. The null hypothesis to be tested is H_o: p = p_o. Find the value of p̂ for this problem, compute the z statistic, and find the P-value. What do you conclude? (A finding of statistical significance in this circumstance does not constitute a proof of discrimination. It can be used, however, to establish a prima facie case. The burden of proof then shifts to the defense.) (Use α = 0.01. Round your test statistic to two decimal places and your P-value to four decimal places.)

Conclusion

Reject the null hypothesis, there is significant evidence that Mexican Americans are underrepresented on juries.

Reject the null hypothesis, there is not significant evidence that Mexican Americans are underrepresented on juries.     

Fail to reject the null hypothesis, there is not significant evidence that Mexican Americans are underrepresented on juries.

Fail to reject the null hypothesis, there is significant evidence that Mexican Americans are underrepresented on juries.

(c) We can reformulate this exercise as a two-sample problem. Here we wish to compare the proportion of Mexican Americans among those selected as jurors with the proportion of Mexican Americans among those not selected as jurors. Let p₁ be the probability that a randomly selected juror is a Mexican American, and let p₂ be the probability that a randomly selected nonjuror is a Mexican American. Find the z statistic and its P-value. (Use α = 0.01. Round your test statistic to two decimal places and your P-value to four

Z _______

Conclusion

Reject the null hypothesis, there is significant evidence of a difference in proportions.

Reject the null hypothesis, there is not significant evidence of a difference in proportions.     

Fail to reject the null hypothesis, there is not significant evidence of a difference in proportions.

Fail to reject the null hypothesis, there is significant evidence of a difference in proportions.

How do your answers compare with your results in (b)?

very different

very similar     

none of the above

You are an analyst working for the life cycle manager of a particular type of cruise missile. Periodically, the inventory of cruise missile engines must be certified and part of the certification requires testing a sample of missile engines then calculating a confidence interval for the true mean flight speed (μ in mph). Based on prior tests, it is appropriate to assume that the missile’s flight speed is normally distributed with known standard deviation, σ=20 mph.

Twenty-five cruise missiles are tested on a range with an average flight speed of?1=375 mph. You are tasked to calculate a 90 percent confidence interval for ?.

Fill in the table to help document your work:

In a consumer study investigating preferences for a new packaging material for granola bars, with the objective of the test being the comparison between the new packaging material (Package A) and the current packaging material (B). A same-difference test was conducted using 180 randomly-selected consumers at a supermarket. Each consumer was presented with two samples, with the combinations being AA, AB, BA or BB. The following results were obtained. A worked example is in your text book (pg. 103)   (7 pts)

Questions:

Question 1: When panelists were presented with the same sample, how many panelists said the samples were the same? (ie. they were correct):

Question 2: When panelists were presented with the different sample, how many panelists said the samples were the same? (ie. they were incorrect):

Question 3: What is the expected total for the “different” condition?

Question 4: What is the calculated value for the same/match condition?

Question 5: What is the calculated value for the different/unmatched condition?

Question 6: What is TOTAL calculated chi-square value?

Question 7: For p=0.05, what is our critical value (taken from Table 19-5)? (0.5 pts)

Question 8: Are there significant differences between these two packaging materials? (0.5 pts)

Question 1: You are evaluating whether the use of red lights during evaluation has a significant effect on liking. Liking was measured as either like or dislike. You collected the below data. Using the Sign Test, did the wine glass significantly influence liking? There is a worked example using the sign test   (5 pts)

Question 1: How many + signs? (0.5 pts)

Question 2: How many 0 values? (0.5 pts)

Question 3: What is the value for N?

Question 4: What is the value for x?

Question 5: Using the table, what is the p value?

Question 6: Does the use of red lights have a significant influence of liking of wine?

Suppose that for a given computer salesperson, the probability distribution of x = the number of systems sold in one month is given by the following table.

(a) Find the mean value of x (the mean number of systems sold).
mean =  

(b) Find the variance and standard deviation of x. (Round the answers to four decimal places.)

(c) What is the probability that the number of systems sold is within 1 standard deviation of its mean value?
P(μ − σ < x < μ + σ ) =  

(d) What is the probability that the number of systems sold is more than 2 standard deviations from the mean?
P(x < μ − 2σ or x > μ + 2σ) =

Two batches of tablets were prepared by two different processes. The potency determinations made on five tablets from each batch were as follows:

• batch A: 5.8, 4.2, 4.6, 5.5, 5.6;
• batch B: 4.5, 4.8, 5.5, 5.2, 4.4.

Test to see if the means of the two batches are equal. Explain how you do that. State your null and alternative hypothesis.

Can you show how to draw the normal curve for each of the problems and label it as well?

Heights of MEN in the U.S. are normally distributed µ = 69.6 inches with σ = 3 inches.

-________ percent (to nearest %) of men in the U.S. are either shorter than 5 ft. or taller than 6 ft?

-In a group of 150 U.S. men, approximately ________ of them should be shorter than 65 inches.

-A male height of _______________ corresponds to the 58th percentile in the U.S. population. -_______________ is the cutoff height to be in the top 12% of male heights in the U.S.

-The middle 72% of U.S. men will be between ________ inches and ________ inches tall. -A man in the U.S. shorter than ___________ inches would be considered "unusually short. ( Can you Show your work or explain answer.)