Question

Let's say I have a bag of marbles. There are 5 different colors: white, black, blue, yellow and red. The claimed ratio of white to black to blue to yellow to red is 1:3:9:1:1. I want to pick a blue marble with closed eyes. I try seven times and get only one blue marble. Is the claimed ratio of blue marbles to other colors overestimated? Use alpha 0.05. Think if you need a 1 or a 2 tail test here.

Answer 1

Solutiion:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: P > 0.60
Alternative hypothesis: P < 0.60

Note that these hypotheses constitute a one-tailed test.

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 = 0.1852
z = (p - P) / S.D

z = - 2.47

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 less than -2.47.

Thus, the P-value = 0.007.

Interpret results. Since the P-value (0.007) is less than the significance level (0.05), we cannot accept the null hypothesis.

Hence we have sufficient evidence in the favor of the claim that ratio of blue marbles to other colors overestimated.