What does it mean to take a sampling of a population? Why do scientists use samplings?

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Self-Test Tutorial

Consider a sample with a mean of 40 and a standard deviation of 5. Use Chebyshev's theorem to determine the percentage of the data within each of the following ranges (to the nearest whole number).

1. 30 to 50, at least _____ %
   
2. 25 to 55, at least _____ %
   
3. 31 to 49, at least _____ %
   
4. 28 to 52, at least _____ %
   
5. 22 to 58, at least _____ %

Suppose the people living in a city have a mean score of 41 and a standard deviation of 8 on a measure of concern about the environment. Assume that these concern scores are normally distributed. Using the ​50% minus −​34% minus −​14% ​figures, approximately what percentage of people have a score​ (a) above 41​, ​(b) above 49​, ​(c) above 25​, ​(d) above 33​, ​(e) below 41​, ​(f) below 49​, ​(g) below 25​, and​ (h) below 33​?

The accompanying data represent the pulse rates​ (beats per​ minute) of nine students. Treat the nine students as a population. Compute the​ z-scores for all the students. Compute the mean and standard deviation of these​ z-scores.

Student   Pulse
Student 1   77
Student 2   60
Student 3   60
Student 4   80
Student 5   72
Student 6   80
Student 7   80
Student 8   68
Student 9   73

Compute the mean of these​ z-scores.

The mean of the​ z-scores is

​(Round to the nearest tenth as​ needed.)

Compute the standard deviation of these​ z-scores.

The standard deviation of the​ z-scores is
(Round to the nearest tenth as​ needed.)

3. Recently, ESPN reported that the average salary of an NFL place-kicker is $1.9 million, while the average salary for an NFL running back is only $1.7 million. This statistic was cited as evidence that, in the current NFL, running backs are considered less valuable than place-kickers. The claim being made based on this statistic is misleading. In what way?   (Hint: Teams have 1 place-kicker and, usually, 5 or 6 running backs, one of whom is their lead back.)

Compute in excel

A college admission officer for an MBA program determines that historically candidates have undergraduate grade averages that are normally distributed with standard deviation of .45. A random sample of 25 applications from the current year yields a sample mean grade point average of 2.90. (i) Find a 95% confidence interval for the population mean, μ. (Round the boundaries to 2 decimal places.) (ii) Based on the same sample results, a statistician computes a confidence interval for the population mean as 2.81< μ < 2.99. Find the α for this interval and the probability content (1- α) as well. (Round to 4 digits.) (Note: the correct α is a higher number than traditional α used; so don’t worry if your number “looks” wrong!) Hint: first calculate α/2 using either the lower bound (2.81) or upper bound (2.99); then calculate α. Finally, calculate the probability content of the interval, which is (1- α). And make sure you use the standard error, not the standard deviation, to calculate α/2.

Consider the following competing hypotheses and accompanying sample data. (You may find it useful to reference the appropriate table: z table or t table)  

H₀: μ₁ – μ₂ = 5
H_A: μ₁ – μ₂ ≠ 5

Assume that the populations are normally distributed with equal variances.

a-1. Calculate the value of the test statistic. (Round intermediate calculations to at least 4 decimal places and final answer to 3 decimal places.)

a-2. Find the p-value.

• p-value < 0.01

• 0.01 ≤ p-value < 0.02

• 0.02 ≤ p-value < 0.05

• 0.05 ≤ p-value < 0.10

• p-value ≥ 0.10

b. At the 5% significance level, can you conclude that the difference between the two means differs from 5?

Consider the following data drawn independently from normally distributed populations: (You may find it useful to reference the appropriate table: z table or t table)

a. Construct the 95% confidence interval for the difference between the population means. Assume the population variances are unknown but equal. (Round all intermediate calculations to at least 4 decimal places and final answers to 2 decimal places.)
  

b. Specify the competing hypotheses in order to determine whether or not the population means differ.
  

• H₀: μ₁ − μ₂ = 0; H_A: μ₁ − μ₂ ≠ 0

• H₀: μ₁ − μ₂ ≥ 0; H_A: μ₁ − μ₂ < 0

• H₀: μ₁ − μ₂ ≤ 0; H_A: μ₁ − μ₂ > 0

c. Using the confidence interval from part a, can you reject the null hypothesis?
  

• Yes, since the confidence interval includes the hypothesized value of 0.

• No, since the confidence interval includes the hypothesized value of 0.

• Yes, since the confidence interval does not include the hypothesized value of 0.

• No, since the confidence interval does not include the hypothesized value of 0.

d. Interpret the results at αα = 0.05.

• We conclude that population mean 1 is greater than population mean 2.

• We cannot conclude that population mean 1 is greater than population mean 2.

• We conclude that the population means differ.

• We cannot conclude that the population means differ.

Consider the following data drawn independently from normally distributed populations: (You may find it useful to reference the appropriate table: z table or t table)

a. Construct the 95% confidence interval for the difference between the population means. (Negative values should be indicated by a minus sign. Round all intermediate calculations to at least 4 decimal places and final answers to 2 decimal places.)

b. Specify the competing hypotheses in order to determine whether or not the population means differ.

• H₀: μ₁ − μ₂ = 0; H_A: μ₁ − μ₂ ≠ 0

• H₀: μ₁ − μ₂ ≥ 0; H_A: μ₁ − μ₂ < 0

• H₀: μ₁ − μ₂ ≤ 0; H_A: μ₁ − μ₂ > 0

c. Using the confidence interval from part a, can you reject the null hypothesis?

• Yes, since the confidence interval includes the hypothesized value of 0.

• No, since the confidence interval includes the hypothesized value of 0.

• Yes, since the confidence interval does not include the hypothesized value of 0.

• No, since the confidence interval does not include the hypothesized value of 0.

d. Interpret the results at   αα = 0.05.

• We conclude that the population means differ.

• We cannot conclude that the population means differ.

• We conclude that population mean 2 is greater than population mean 1.

• We cannot conclude that population mean 2 is greater than population mean 1.

Please provide an example and then discuss how regression analysis may be used as a forecasting tool.

Thank you.

Indicate whether each measure is a nominal, ordinal, or interval/ratio measure. Explain the reasons for your choices:

a. inches in a yardstick:

b. Virginia driver’s license customer ID:

c. dollars as a measure of income:

d. order of finish in a horse race:

e. intelligence test scores:

Longwave (LW) and shortwave (SW) are two appearance measures used in the automotive industry to rate the quality of a paint job. These two measures are generally related. Values for 13 cars are given. Use these data to answer the following:

What value represents the coefficient of correlation between the LW and SW appearances?

Group of answer choices

0.8199

0.6723

0.3461

0.124

It is thought that prehistoric Indians did not take their best tools, pottery, and household items when they visited higher elevations for their summer camps. It is hypothesized that archaeological sites tend to lose their cultural identity and specific cultural affiliation as the elevation of the site increases. Let x be the elevation (in thousands of feet) of an archaeological site in the southwestern United States. Let y be the percentage of unidentified artifacts (no specific cultural affiliation) at a given elevation. The following data were obtained for a collection of archaeological sites in New Mexico. x 5.17 5.87 6.25 6.75 7.25 y 20 11 33 37 62 Complete parts (a) through (e), given Σx = 31.29, Σy = 163, Σx2 = 198.3733, Σy2 = 6823, Σxy = 1073.47, and r ≈ 0.859. (a) Draw a scatter diagram displaying the data. Get Flash Player Flash Player version 10 or higher is required for this question. You can get Flash Player free from Adobe's website. (b) Verify the given sums Σx, Σy, Σx2, Σy2, Σxy, and the value of the sample correlation coefficient r. (Round your value for r to three decimal places.) Σx = Σy = Σx2 = Σy2 = Σxy = r = (c) Find x, and y. Then find the equation of the least-squares line y hat = a + bx. (Round your answers for x and y to two decimal places. Round your answers for a and b to three decimal places.) x = y = y hat = + x (d) Graph the least-squares line. Be sure to plot the point (x, y) as a point on the line. WebAssign Plot WebAssign Plot WebAssign Plot WebAssign Plot (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? (Round your answer for r2 to three decimal places. Round your answers for the percentages to one decimal place.) r2 = explained % unexplained % (f) At an archaeological site with elevation 6.1 (thousand feet), what does the least-squares equation forecast for y = percentage of culturally unidentified artifacts? (Round your answer to two decimal places.) %

The accompanying data represent the miles per gallon of a random sample of cars with a​ three-cylinder, 1.0 liter engine.

32.4
34.1
34.5
35.7
36.1
36.3
37.5
37.7
37.9
38.1
38.3
38.5
38.7
39.1
39.5
39.8
39.9
40.6
41.3
41.6
42.3
42.7
43.8
49.0

Find the following:

A) 5! =

B) ₇C₂ =

C) ₇P₂ =