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Chapter 10: Problem 64

Suppose there is a theory that \(90 \%\) of the people in the United States dream in color. You survey a random sample of 200 people; 198 report that they dream in color, and 2 report that they do not. You wish to verify the claim made in the theory.

Short Answer

Expert verified
After gathering all the data, the calculation shall reveal if there's enough evidence to reject or fail to reject the null hypothesis. P-value will provide conclusive evidence.

Step by step solution

01

Identify Null and Alternative Hypothesis

The Null Hypothesis (\(H_0\)) is the theory's claim: 90% people dream in color. Alternative Hypothesis (\(H_1\)) will be: Not 90% people dream in color.
02

Calculate Sample Proportion

Sample proportion (\(p\)) is the proportion of people in the sample who dream in color. It is calculated as \(p = \frac{x}{n}\), where \(x\) is number of people who dream in color (198) and \(n\) is total number of sample (200). So, \(p = \frac{198}{200} = 0.99.\)
03

Check Conditions

Before proceeding, whether the conditions for a z-proportion test are met needs to be ascertained: (1) The sampling method is simple random sampling. It is stated in the problem, so this condition is met. (2) The sample size should be large enough: \(np \geq 10\) and \(n(1-p) \geq 10\). In this case, \(n = 200, p = 0.90\), so \(np = 180\) and \(n(1-p) = 20\) which are both greater than 10. So, this condition is also met.
04

Perform Z-Test

The z-score is a measure of how many standard deviations an observation or datum is from the mean. It's calculated as \( Z = \frac{p - P}{\sqrt{\frac{P(1-P)}{n}}}\), where \(p\) is a sample proportion, \(P\) is a population proportion (0.90 in this case) and \(n\) is a sample size (200 in this case). Substituting the values will give the z-score.
05

Determine P-Value and make a conclusion

Find the p-value associated with obtained z-score. If the p-value is less than the level of significance (0.05 for 95% confidence level), reject the null hypothesis and conclude that the alternative hypothesis is true. If the p-value is not less than the level of significance, there is sufficient evidence to support the null hypothesis.

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Most popular questions from this chapter

Rats had a choice of freeing another rat or eating chocolate by themselves. Most of the rats freed the other rat and then shared the chocolate with it. The table shows the data concerning the gender of the rat in control. $$\begin{array}{lcc} & \text { Male } & \text { Female } \\\\\hline \text { Freed Rat } & 17 & 6 \\\\\text { Did not } & 7 & 0 \\ \hline\end{array}$$ a. Can a chi-square test for homogeneity or independence be performed with this data set? Why or why not? b. Determine whether the sex of a rat influences whether or not it frees another rat using a significance level of \(0.05\).

Suppose you are interested in whether more than \(50 \%\) of voters in California support a proposition (Prop X). After the vote. you find the total number that support it and the total number that oppose it.

Fill in the blank by choosing one of the options given: Chi-square goodness-of-fit data are often summarized with _________ (one row or one column of observed counts - but not both, or at least two rows and at least two columns of observed counts).

According to a 2017 report, \(64 \%\) of college graduate in Illinois had student loans. Suppose a random sample of 80 college graduates in Illinois is selected and 48 of them had student loans. (Source: Lendedu.com) a. What is the observed frequency of college graduates in the sample who had student loans? b. What is the observed proportion of college graduates in the sample who had student loans? c. What is the expected number of college graduates in the sample to have student loans if \(64 \%\) is the correct rate? Do not round off.

Refer to the description in exercise 10.71. There were 22 trials with only cockroaches (no robots) that went under one shelter. In 16 of these 22 trials, the group chose the darker shelter, and in 6 of the 22 the group chose the lighter shelter. There were 28 trials with a mixture of real cockroaches and robots that all went under one shelter. In 11 of these trials, the group chose the darker shelter, and in 17 the group chose the lighter shelter. The robot cockroaches were programmed to choose the lighter shelter (as well as preferring groups; Halloy et al. 2007 ) Is the introduction of robot cockroaches associated with the type of shelter when the group went under one shelter? Assume cockroaches were randomly sampled from some meaningful population of cockroaches. a. Use the chi-square test to see whether the presence or absence of robots is associated with whether they went under the darker or the brighter shelter. Use a significance level of \(0.05\) b. Do Fisher's Exact Test with the data. If your software does not do Fisher's Exact Test, search the Internet for a Fisher's Exact Test calculator and use it. Report the p-value and your conclusion. c. Compare the p-values for parts a and b. Which do you think is the more accurate procedure? The p-values that result from the two methods in this question are closer than the p-values in the previous question. Why do you think that is?
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