Question

Question text

It has been theorized that senior citizens dream in black and white at a higher rate than the rest of the population because they were heavily exposed to black and white media in movies, television, and print. A study was conducted to determine the proportion of people who dream in black and white instead of color. Among 306 people over the age of 55, 68 dream in black and white, and among 298 people under the age of 25, 13 dream in black and white (based on data from “Do We Dream in Color?” by Eva Murzyn, Consciousness and Cognition, Vol. 17, No. 4). We want to use a 0.01 significance level to see if we can support the claim that people over 55 dream in black and white at a higher rate than people under 25.

1.Before we conduct a hypothesis test, we should check if the claim we’re testing is even plausible. What is the value of pp for people under 25, written as a decimal between 0 and 1, out to three decimal places?

2.What is the value of pp for people over 55, written as a decimal between 0 and 1, out to three decimal places?

3.What is the value of the test statistic, written to two decimal places?

Answer 1

1). 68 out of 306 people aged above 55 dream in black and white

Hence p1 = proportion of people aged above 55 who dream in black and white

                = 68/306 = 0.2222

           p1 = 0.2222

2). 13 out of 298 people under the age of 25 dream in black and white

Hence p2 = proportion of under the age of 25 who dream in black and white

              = 13/298 = 0.0436

           p2 = 0.0436

Since the number of people who dream in black and white is more than 10 in both the groups, we can conduct the hypothesis test

3)

	n	Proportion (p)
above 55 years	n1 = 306	p1 = 0.2222
under 25 years	n2 = 298	p2 = 0.0436

The null and alternative hypotheses are
Ho :  P1 = P2
Ha :  P1 > P2
where P1, P2 are the population proportions of people who dream in black and white for groups above 55 years of age and below 25 years respectively

α = 0.01                                              ...Level of significance
Using the given formulae, we get

Pooled Proportion

p̂ = 0.1341

Standard Error (SE)

SE = 0.0277

z-statistic

z-statistic = 6.44

P-value
For z = 6.44 we find the Greater Than p-value using z tables or Excel function NORM.DIST
p-value = 1 - NORM.S.DIST(6.44, TRUE)
p-value = 0

Decision
0 < 0.01
that is p-value is less than alpha.
Hence we Reject Ho.

Conclusion
There exists enough statistical evidence at α = 0.01 to show that senior citizens dream in black and white at a higher rate than the rest of the population