Suppose you want to conduct a survey of a sample of registered at the University. Discuss some techniques that would be appropriate to select the sample. Discuss the advantages and disadvantages of each.

In the Focus Problem at the beginning of this chapter, a study was described comparing the hatch ratios of wood duck nesting boxes. Group I nesting boxes were well separated from each other and well hidden by available brush. There were a total of 487 eggs in group I boxes, of which a field count showed about 270 hatched. Group II nesting boxes were placed in highly visible locations and grouped closely together. There were a total of 808 eggs in group II boxes, of which a field count showed about 262 hatched.

(a) Find a point estimate p̂₁ for p₁, the proportion of eggs that hatch in group I nest box placements. (Round your answer to three decimal places.)
p̂₁ =  

Find a 95% confidence interval for p₁. (Round your answers to three decimal places.)

(b) Find a point estimate p̂₂ for p₂, the proportion of eggs that hatch in group II nest box placements. (Round your answer to three decimal places.)
p̂₂ =  

Find a 95% confidence interval for p₂. (Round your answers to three decimal places.)

(c) Find a 95% confidence interval for p₁ − p₂. (Round your answers to three decimal places.)

Does the interval indicate that the proportion of eggs hatched from group I nest boxes is higher than, lower than, or equal to the proportion of eggs hatched from group II nest boxes?

Because the interval contains only negative numbers, we can say that a higher proportion of eggs hatched in highly visible, closely grouped nesting boxes.We can not make any conclusions using this confidence interval.     Because the interval contains only positive numbers, we can say that a higher proportion of eggs hatched in well-separated and well-hidden nesting boxes.Because the interval contains both positive and negative numbers, we can not say that a higher proportion of eggs hatched in well-separated and well-hidden nesting boxes.

(d) What conclusions about placement of nest boxes can be drawn? In the article discussed in the Focus Problem, additional concerns are raised about the higher cost of placing and maintaining group I nest box placements. Also at issue is the cost efficiency per successful wood duck hatch.

A greater proportion of wood duck eggs hatch if the eggs are laid in highly visible, closely grouped nesting boxes.No conclusion can be made.     A greater proportion of wood duck eggs hatch if the eggs are laid in well-separated, well-hidden nesting boxes.The eggs hatch equally well in both conditions.

Economists often look at retail sales data to gauge the state of the economy. This is especially so in a recession year, when consumer spending has decreased. Consider the following table, which shows U.S. monthly nominal retail sales for 2009. Sales are measured in millions of dollars and have been seasonally adjusted. Also included in the table is the corresponding producer price index (PPI) for 2009. Month Sales PPI Month Sales PPI January 339,453 171.3 July 341,568 172.0 February 341,371 171.1 August 349,899 174.3 March 338,205 169.6 September 342,752 173.3 April 337,389 170.8 October 346,691 174.0 May 338,934 170.8 November 353,535 176.4 June 341,992 173.9 December 352,858 177.3 a. How many times were nominal sales below that of the previous month? b-1. Use the PPI to compute sales in real terms. (Round your answers to 2 decimal places.) b-1. Use the PPI to compute sales in real terms. (Round your answers to 2 decimal places.)

month. real sales

How many times were real sales below that of the previous month?

Compute the total percentage increase in nominal as well as real retail sales in 2009. (Round your answers to 2 decimal places.)

Can economists feel optimistic about the economy based on the retail sales data?

Use the standard normal table to find the​ z-score that corresponds to the cumulative area 0.0188. If the area is not in the​ table, use the entry closest to the area. If the area is halfway between two​ entries, use the​ z-score halfway between the corresponding​ z-scores.

An engineer is going to redesign an ejection seat for an airplane. The seat was designed for pilots weighing between 120 lb and 161 lb. The new population of pilots has normally distributed weights with a mean of 129 lb and a standard deviation of 32.7 lb. a. If a pilot is randomly​ selected, find the probability that his weight is between 120 lb and 161lb. The probability is approximately _____ (Round to four decimal places as​ needed.) b. If 32 different pilots are randomly​ selected, find the probability that their mean weight is between 120 lb and 161 lb. The probability is approximately ____. ​(Round to four decimal places as​ needed.) c. When redesigning the ejection​ seat, which probability is more​ relevant?

How does one look up the z table

Consider babies born in the "normal" range of 37–43 weeks gestational age. A paper suggests that a normal distribution with mean μ = 3500 grams and standard deviation σ = 619 grams is a reasonable model for the probability distribution of the continuous numerical variable x = birth weight of a randomly selected full-term baby.

a. What is the probability that the birth weight of a randomly selected full-term baby is either less than 2000 g or greater than 5000 g? (Round your answer to four decimal places.)

b How would you characterize the most extreme 0.1% of all full-term baby birth weights? (Round your answers to the nearest whole number.) The most extreme 0.1% of birth weights consist of those greater than ____ grams and those less than ____ grams.

c. If x is a random variable with a normal distribution and a is a numerical constant (a ≠ 0), then y = ax also has a normal distribution. Use this formula to determine the distribution of full-term baby birth weight expressed in pounds (shape, mean, and standard deviation), and then recalculate the probability from 0.700. (Round your answer to four decimal places.)

Twelve runners are asked to run a 10-kilometer race on each of two consecutive weeks. In
one of the races, the runners wear one brand of shoe and in the other a different brand. The
brand of shoe they wear in which race is determined at random. All runners are timed and
are asked to run their best in each race. The results (in minutes) are given below:
Runner Brand 1 Brand 2
1 31.23 32.02
2 29.33 28.98
3 30.50 30.63
4 32.20 32.67
5 33.08 32.95
6 31.52 31.53
7 30.68 30.83
8 31.05 31.10
9 33.00 33.12
10 29.67 29.50
11 30.55 30.57
12 32.12 32.20
Use the sign test for matched pairs to determine if there is evidence that times using Brand
1 tend to be faster than times using Brand 2.
(a) What are the hypotheses we wish to test?
i. H0 : μ = 0 versus Ha : μ > 0, where μ =the mean of the differences in running
times (Brand 2-Brand 1) for all runners who run this race twice wearing the two
brands of shoes.
ii. H0 : p = 1
2 versus Ha : p , 1
2 , where p = the proportion of running times using
Brand 1 that are faster than times using Brand 2.
iii. H0 : p = 1
2 versus Ha : p >
1
2 , where p = the proportion of running times using
Brand 1 that are faster than times using Brand 2.
iv. H0: population median =0 versus Ha: population median , 0, where the median
of the differences in running times for all runners who run this race twice wearing
the two brands of shoes is measured for Brand 2-Brand 1.
(b) What is the (approximate) value of the P-value?
(c) Determine which of the following statements is true.
i. We would not reject the null hypothesis of no difference at the 0.10 level.
ii. We would reject the null hypothesis of no difference at the 0.10 level but not at the
0.05 level.
iii. We would reject the null hypothesis of no difference at the 0.05 level but not at the
0.01 level.
iv. We would reject the null hypothesis of no difference at the 0.01 level.

According to a government energy agency, the mean monthly household electricity bill in the United States in 2011 was $108.99 . Assume the amounts are normally distributed with standard deviation $19.00 .

(a) What proportion of bills are greater than $133 ?

(b) What proportion of bills are between $82 and $140 ?

(c) What is the probability that a randomly selected household had a monthly bill less than $119 ? Round the answers to at least four decimal places.

1. Randomly selected statistics students participated in an experiment to test their ability to determine when 1 min has passed.  Forty students yielded a sample mean of 58.3 sec.  Assuming that s = 9.5 sec, use a 0.05 level of significance to test the claim that the population mean is equal to 60 sec.  Be sure and include the appropriate interval (for alpha = .05, it is a 95% confidence level) in your conclusion. Conduct a hypothesis test.

3. Assume we want to test the claim that the mean attendance in the arena is more than 20,000 fans.  In order to test this we selected 8 concerts at random and took the attendance from each concert. The sample average from those 8 concert was 20,450.  If we assume that we know the sample standard deviation is 500 fans.  Show all 6 steps in conducting a hypothesis test to test this claim as a normal distrubution. Include the appropriate confidence interval in the conclusion.

b. Conduct the test again showing all 6 steps accept assume now that we know sigma and that the value given is s = 500 fans .   

5. Assume the coach wanted a reliable estimate of her ability to play and allowed him to hit 10 rounds and took the average number of strokes of those 10 rounds.  His rounds were:

72             74           69           72           71           72            71            72            72            73

Is there sufficient evidence to suggest that her true average is less than 73?  Perform the 6 step hypothesis test at the alpha = .01 level of significance to find out!  You may assume that her round scores come from a normal distribution.   Be sure to include the appropriate confidence interval (the one that will always “agree” with your test) in your conclusion.

This problem demonstrates a possible (though rare) situation that can occur with group comparisons. The groups are sections and the dependent variable is an exam score. Section 1 77.6 69.1 53.5 68.2 73.9 63.3 61.3 71.9 60.2 Section 2 58.5 61.3 57.5 34.5 73.2 37.1 50.3 51.2 76.8 Section 3 67.6 63.7 60.5 71.7 65 66.8 58.5 74.2 68.2 Run a one-factor ANOVA (fixed effect) with α = 0.05 α = 0.05 . Report the F-ratio to 3 decimal places and the P-value to 4 decimal places. F = F = p = p = What is the conclusion from the ANOVA? reject the null hypothesis: at least one of the group means is different fail to reject the null hypothesis: not enough evidence to suggest the group means are different Calculate the group means for each section: Section 1: M 1 = M 1 = Section 2: M 2 = M 2 = Section 3: M 3 = M 3 = Report means accurate to 2 decimal places. Conduct 3 independent sample t-tests for each possible pair of sections. (Though we will see later that it might not be appropriate, retain the significance level α = 0.05 α = 0.05 .) Report the P-value (accurate to 4 decimal places) for each pairwise comparison. Compare sections 1 & 2: p = p = Compare sections 1 & 3: p = p = Compare sections 2 & 3: p = p = Based on these comparisons, which pair of groups have statistically significantly different means? group 1 mean is statistically different from group 2 mean group 1 mean is statistically different from group 3 mean group 2 mean is statistically different from group 3 mean none of the group means are statistically significantly different from each other Thought for reflection: What do the results of the pairwise comparisons suggest about the original conclusion from the ANOVA?

Investigators are interested in whether a new diet lowers total cholesterol in a group of individuals. They take a sample of 28 participants on the new diet. They know that the mean cholesterol level of the general population (µ) is 201 mg/dL, but they do not have information on the standard deviation of the population. The mean total cholesterol level in their sample of participants is 195 mg/dL and their sample has a standard deviation of 9 mg/dL.

Your research question of interest is:

Does the population of people on a new diet have a different mean cholesterol level than the general population?

You will use the information above to complete the question parts below for a one sample t-test.

Part A: Which of the following represents the appropriate null hypothesis (H₀), given this research question of interest?

Part B: Which of the following represents the appropriate alternative hypothesis (H₁), given this research question of interest?

Part C: Which of the following represents the appropriate critical values, testing at an alpha level (α) of 0.05?

Part D: What is the t-statistic associated with your test? (rounded to the nearest hundredth)

Part E: Given your test results, what is your decision about the null hypothesis?

Part F: The best interpretation of the appropriate decision regarding the null hypothesis would be: "Based on our study, we                            [ Select ]                       ["have", "do NOT have"]         enough evidence to conclude that the population of people on a new diet has a different mean cholesterol level than the general population."

Part G: Which of the following represents a 95% confidence interval for the unknown population mean cholesterol level of people on a new diet.

(hint: since the population standard deviation (σ) is unknown, you must use the t-distribution for your confidence interval construction.)

Which of the below represents the best interpretation of the 95% confidence interval you calculated on the previous question for the unknown population mean cholesterol level of people on the new diet?

*note: the actual values for the confidence interval limits are left blank intentionally

a. About 95% of people on the new diet have a cholesterol level between ___ and ___.

b. We are 95% confident that the unknown population mean cholesterol level of people on the new diet falls between ___ and ___.

c. It is guaranteed that the unknown population mean cholesterol level of people on the new diet falls between ___ and ___.

d. We are 95% confident that the sample mean cholesterol level of people on the new diet falls between ___ and ___.

Describe a research situation where survival analysis would be an appropriate technique

Find the probability that a randomly chosen group of 9 people have no birthdays in common, but randomly chosen person #10 has a birthday in common with one of the first 9.

You measure 21 textbooks' weights, and find they have a mean weight of 46 ounces. Assume the population standard deviation is 8.1 ounces. Based on this, construct a 95% confidence interval for the true population mean textbook weight.

Give your answers as decimals, to two places

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