Question

John purchases a bag of marbles and observes and counts their colors. The bag contains 142 marbles of which 11 are orange. Use the p-value method and a 7% significance level to test the claim that the percentage of orange marbles is at most 11%. What population parameter is being tested? How many populations are being tested? Calculate the sample proportion (rounded to the nearest ten-thousandth). What is the claim? The claim is the _________ hypothesis. What is the alternative hypothesis? What is the test statistic (rounded to the nearest hundredth)? The critical region is best described as ____________. What is the p-value (rounded to the nearest ten-thousandth)? What is the statistical conclusion? What is the wordy conclusion?

Answer 1

The population parameter is population proportion.

Number of populations = 142 * 100/5 = 2840

The sample proportion() = 11/142 = 0.0775

Claim : H₀: P < 0.11

The claim is the null hypothesis.

H_a: P > 0.11

The test statistic z = ( - P)/sqrt(P(1 - P)/n)

                             = (0.0775 - 0.11)/sqrt(0.11 * (1 - 0.11)/142)

                             = -1.24

At alpha = 0.07, the critical value is z_0.93 = 1.48

Reject H₀, if z > 1.48

P-value = P(Z > -1.24)

             = 1 - P(Z < -1.24)

             = 1 - 0.1075

             = 0.8925

Since the P-value is greater than the significance level (0.8925 > 0.07), so we should not reject the null hypothesis.

So at 7% significance level there is sufficient evidence to support the claim that the percentage of orange marbles is at most 11%.