When entering the city of Cityville, West Virginia you will come across a sign t

Question

When entering the city of Cityville, West Virginia you will come across a sign that states: “Welcome to Cityville– The Friendly City.” A social psychologist was interested in determining if Cityville really was “The Friendly City.” She obtained a sample of 105 Cityville residents, and had them fill out a questionnaire concerning friendliness. In the general population, the mean friendliness score is 30 with a standard deviation of 3. The mean of the Cityville sample was 31.

a.                  Find the probability of obtaining such a sample mean from the aforementioned population.

b.                  Calculate the 95% confidence interval.

c.                  Calculate the effect size.

d.                  Describe the results of the study, indicating whether or not this is a statistically significant results, what the confidence interval tells us, what the effect size tells us, and what we can conclude about the friendliness of Cityville residents in relation to the general population.

Explanation / Answer

(A)
P(X=31) = P(30.5<X<31.5)
P(X<30.5) = P(z<(30.5-30)/(3/sqrt(105))) = P(z<1.7078) = 0.9562

P(X<31.5) = P(z<(31.5-30)/(3/sqrt(105))) = P(z<5.1235) = 1

P(X=31) = P(30.5<X<31.5) = P(X<31.5) - P(X<30.5) = 1 - 0.9562 = 0.0438

(B) z-score for 95% CI, z = 1.96
Margin of Error (ME) = z*sigma/sqrt(n) = 1.96*3/sqrt(105) = 0.5738

Confidence Interval = (31 - 0.5738, 31 + 0.5738) = (30.4262, 31.5738)

(C) Effect size = (31 - 30)/3 = 1/3 = 0.3333

(D)
The results are statistically significanct which indicates the friendliness of Cityville is not less than 31.

An effect size is exactly equivalent to a 'Z-score' of a standard Normal distribution. An effect size of 0.33 means that the score of the average person in the experimental group(sample) is 0.33 standard deviations above the average person in the control group(population), and hence exceeds the scores of 32% of the control group. This means friendliness of Cityville exceeds the friendliness of the general population.

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