3. (10 pts) Do we dream in color? In the 1940s, before the age of television, co
ID: 2946693 • Letter: 3
Question
3. (10 pts) Do we dream in color? In the 1940s, before the age of television, color movies, and video games, 29% of the American population reported dreaming in color. A psychologist suspects that the present-day proportion might be higher, now that we are surrounded with color imagery. In a random sample of 113 people, 92 reported dreaming in color (Schwitzgebel 2003). Conduct a hypothesis test to determine if the psychologist's suspicion is correct. In the context of this problem, what would it mean if we made a Type 1 error? What is the probability of making this kind of error? a) Clearly show all 7 steps. b)Explanation / Answer
SOlution:-
a)
State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.
Null hypothesis: P < 0.29
Alternative hypothesis: P > 0.29
Note that these hypotheses constitute a one-tailed test. The null hypothesis will be rejected only if the sample proportion is too small.
Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method, shown in the next section, is a one-sample z-test.
Analyze sample data. Using sample data, we calculate the standard deviation (S.D) and compute the z-score test statistic (z).
S.D = sqrt[ P * ( 1 - P ) / n ]
S.D = 0.04269
z = (p - P) / S.D
z = 12.28
where P is the hypothesized value of population proportion in the null hypothesis, p is the sample proportion, and n is the sample size.
Since we have a one-tailed test, the P-value is the probability that the z-score is gerater than 12.28.
Thus, the P-value = less than 0.0001.
Interpret results. Since the P-value (almost 0) is less than the significance level (0.05), we have to accept the null hypothesis.
b) In this context the type I error means that we are concluding that present day proportion is higher while actually it is not higher than 0.29. The probability of making a type I error is 0.05.
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