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Specify the differences between a large-sample and a small-sample test of a hypothesis about a population mean m. Focus on the assumptions and test statistics.

Short Answer

Expert verified

A large-sample, as well as a small-sample hypothesis test of a population, mean, μvary because a large sample can be considered to be typically disturbed.

Step by step solution

01

Hypothesis

A hypothesis is an assumption formed based on facts. This is the first step in every inquiry that converts research issues into forecasts. It consists of variables, a population, as well as the relationship among the variables. A research hypothesis is a theory used to evaluate the link among two or more variables.

02

Explanation

In a population's large-sample and small-sample hypothesis tests, the mean varies because a large sample can be considered typically disturbed. As a result, one can use a normally distributed area to conduct an extensive sample hypothesis test. However, this assumption may not remain valid for a smaller sample. Therefore we utilize the t distribution for a minor sample hypothesis test.

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Most popular questions from this chapter

Factors that inhibit learning in marketing. What factors inhibit the learning process in the classroom? To answer this question, researchers at Murray State University surveyed 40 students from a senior-level marketing class (Marketing Education Review). Each student was given a list of factors and asked to rate the extent to which each factor inhibited the learning process in courses offered in their department. A 7-point rating scale was used, where 1 = “not at all” and 7 = “to a great extent.” The factor with the highest rating was instructor related: “Professors who place too much emphasis on a single right answer rather than overall thinking and creative ideas.” Summary statistics for the student ratings of this factor are\(\overline x = 4.70\),\(s = 1.62\)

a. Conduct a test to determine if the true mean rating for this instructor-related factor exceeds 4. Use\(\alpha = 0.05\).Interpret the test results.

b. Examine the results of the study from a practical view, and then discuss why “statistically significant” does not always imply “practically significant.”

c. Because the variable of interest, rating, is measured on a 7-point scale, it is unlikely that the population of ratings will be normally distributed. Consequently, some analysts may perceive the test, part a, to be invalid and search for alternative methods of analysis. Defend or refute this argument

A random sample of n observations is selected from a normal population to test the null hypothesis that µ=10.Specify the rejection region for each of the following combinations of \(Ha,\alpha ,\) and n:

a.\(Ha:\)µ\( \ne 10;\alpha = .05.;n = 14\)

b.\(Ha:\)µ\( > 10;\alpha = .01;n = 24\)\(\)

c.\(Ha:\)µ\( > 10;\alpha = .10;n = 9\)

d.\(Ha:\)µ <\(10:\alpha = .01;n = 12\)

e.\(Ha:\)µ\( \ne 10;\alpha = .10;n = 20\)

f. \(Ha:\)µ<\(10;\alpha = .05;n = 4\)

Crude oil biodegradation. Refer to the Journal of Petroleum Geology (April 2010) study of the environmental factors associated with biodegradation in crude oil reservoirs, Exercise 6.38 (p. 350). Recall that 16 water specimens were randomly selected from various locations in a reservoir on the floor of a mine and that the amount of dioxide (milligrams/liter)—a measure of biodegradation—as well as presence of oil were determined for each specimen. These data are reproduced in the accompanying table.

a. Conduct a test to determine if the true mean amount of dioxide present in water specimens that contained oil was less than 3 milligrams/liter. Use\(\alpha = .10\)

Coffee markets that conform to organic standards focus on the environmental aspects of coffee growing, such as the use of shade trees and a reduced reliance on chemical pesticides. A study of organic coffee growers was published in Food Policy (Vol. 36, 2010). In a representative sample of 845 coffee growers from southern Mexico, 417 growers were certified to sell to organic coffee markets while 77 growers were transitioning to become organic certified. In the United States, 60% of coffee growers are organic certified. Is there evidence to indicate that fewer than 60% of the coffee growers in southern Mexico are either organic certified or transitioning to become organic certified? State your conclusion so that there is only a 5% chance of making a Type I error.

Consider the test \({H_0}:\mu = 70\) versus \({H_a}:\mu \ne 70\) using a large sample of size n = 400. Assume\(\sigma = 20\).

a. Describe the sampling distribution of\(\bar x\).

b. Find the value of the test statistic if\(\bar x = 72.5\).

c. Refer to part b. Find the p-value of the test.

d. Find the rejection region of the test for\(\alpha = 0.01\).

e. Refer to parts c and d. Use the p-value approach to

make the appropriate conclusion.

f. Repeat part e, but use the rejection region approach.

g. Do the conclusions, parts e and f, agree?

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