Question

Suppose that in a random selection of 100 colored​ candies, 30​% of them are blue. The candy company claims that the percentage of blue candies is equal to 26​%. Use a 0.05 significance level to test that claim.

Answer 1

To assess the claim that 26% of candies are blue, a hypothesis test for a single population proportion is performed: stating hypotheses, calculating the sample proportion, standard error, z-test statistic, comparing to the critical value, and concluding if the claim is supported.

Step-by-step explanation:

To test the candy company's claim that 26% of their candies are blue at a 0.05 significance level, we can perform a hypothesis test for a single population proportion. Here are the steps:

1. State the null hypothesis H0: p = 0.26, which is the company's claimed proportion of blue candies, and the alternative hypothesis Ha: p ≠ 0.26, which suggests the proportion of blue candies is not 26%.
2. Find the sample proportion (p') of blue candies. In this case, p' = 0.30 since 30% of the 100 candies sampled are blue.
3. Use the standard error of the proportion SE = sqrt(p(1 - p) / n), where p is the hypothesized proportion, and n is the sample size. SE = sqrt(0.26 * 0.74 / 100).
4. Calculate the test statistic using the formula Z = (p' - p) / SE.
5. Compare the Z-value to the critical value from the standard normal distribution for a two-tailed test at α = 0.05. If the absolute value of Z is greater than the critical value, reject H0.
6. Conclude whether the company's claim is supported or not based on the hypothesis test.

It's important to note the use of a two-tailed test due to the alternative hypothesis implying that the proportion may be higher or lower than 26%, rather than one specific direction.