MATH 215 Introduction to Statistics Assignment 5

Overview

This assignment covers content from Unit 5. It assesses your ability to use sampling distributions in hypothesis testing about the difference between two or more population means or the difference between two population proportions, including tests for experiments with more than two categories and tests about contingency tables.

Instructions

•  Show all your work and justify all of your answers and conclusions.
• Keep your work to 4 decimals, unless otherwise stated.
•   Note: Finishing a test of hypotheses with a statement like “reject ” or “do not reject ” will be insufficient for full marks. You must also provide a written concluding statement in the context of the problem itself. For example, if you are testing hypotheses about the effectiveness of a medical treatment, you must conclude with a statement like, “we can conclude that the treatment is effective” or “we cannot conclude that the treatment is effective.”

1) A researcher is interested in examining the cholesterol levels of heart-attack patients. Cholesterol levels are measured for 28 heart-attack patients (2 days after their attacks) and 30 other hospital patients who did not have a heart attack. The researcher believes that cholesterol levels will be higher for the heart-attack patients. 

Assume that the cholesterol levels for both populations are normally distributed and that the population standard deviations are equal.

Using a 5% significance level, can the researcher conclude that the mean cholesterol level of heart-attack patients is greater than the mean cholesterol level of non-heart-attack patients? Formulate and test the appropriate hypotheses. State and explain your conclusion within the context of the question. Use the critical value approach. 

2) A manufacturer wanted to improve on the processes used to produce electrical components. At the beginning of the year, the factory randomly examined 9,000 electrical components, and of these a total of 900 components were rejected after a quality-control inspection. A project was deployed to fix the problem. Following the project, 7,000 components were randomly picked to be examined. Of these, a total of 600 were rejected. Did the project intervention improve the process?

Test at the 2% significance level whether the population proportion of rejected components decreased after the project compared to the population proportion prior to the project. Formulate and test the appropriate hypotheses. Use the p-value approach. Be sure to clearly state and explain your conclusion within the context of the question.

3) Researchers counted the number of breeding sea turtles on various sections of beach property in Cancun every year. Nine randomly selected sections of beach were used. The following table shows the number of counted sea turtles for two successive years (2015 and 2016).

At the 5% significance level, can it be concluded that the number of breeding sea turtles in 2015 is different from the number of sea turtles in 2016? Formulate and test the appropriate hypotheses. Use the critical value approach. Assume the population of paired differences has a normal distribution. Clearly state and explain your conclusion within the context of the question.

4) After introducing a new teaching curriculum, a teacher is interested in whether the grade distribution in his course is significantly different than it was in previous years. The distribution of grades before the introduction of the new curriculum was as follows:

A random sample of 150 students taken after the introduction of the new curriculum provided the following results:

Does the observed data contradict the hypothesis? Formulate and test the appropriate hypotheses at the 1% significance level. Use the critical value approach. Clearly state and explain your conclusion within the context of the problem

5) A marketing firm that markets refrigerators is interested in studying consumer behavior in the context of purchasing a particular brand of refrigerator. It wants to know, in particular, whether the income-level of the consumers influences their choice of refrigerator brand. Currently, there are three brands available in the marketplace. Brand A is a premium brand, Brand B is a more moderately priced brand, and Brand C is the most economical brand.

A representative stratified random sampling procedure was adopted covering the entire market using income as the basis of selection. Income was classified into three categories: lower, middle and high. A sample of 200 consumers participated in this study and produced the following data:

At the 5% significance level, can it be concluded that there is a relationship between income-level and brand preference? Formulate and test the appropriate hypotheses. Use the critical value approach. Clearly state and explain your conclusion within the context of the question.

6) Three colors of warning lights can be used on an automobile instrument panel. A researcher was interested to know whether users would have different reaction times depending on the color used in the panel. To find out, she randomly assigned, from 15 participants in total, 5 participants to each one of the 3 colors, and then measured their reaction times (in hundredths of a second, with decimal points deleted). The following data were obtained:

Given that the necessary assumptions are satisfied, can it be concluded, at the 5% level of significance, that not all mean reaction times to the colors are equal? Formulate and test the appropriate hypotheses. Use the critical value approach. Clearly state and explain your conclusion within the context of the question.

7) The following ANOVA table is based on information obtained for five samples selected from five independent populations that are normally distributed with equal variances:

1. Fill in the missing values in the table as indicated by the blanks (---).
2. Using a significance level of  , indicate what your null and alternative hypotheses would be in this situation. Test these hypotheses, state your conclusion and explain its meaning in the context of this problem.