Answer: To test the candy company's claim that the percentage of blue candies is equal to 22%, we can conduct a hypothesis test using the 0.01 significance level (also known as the alpha level or level of significance).
Let's set up the null and alternative hypotheses:
Null Hypothesis (H0): The percentage of blue candies is equal to 22%.
Alternative Hypothesis (Ha): The percentage of blue candies is not equal to 22%.
Mathematically, the hypotheses can be expressed as:
H0: p = 0.22
Ha: p ≠ 0.22
where p represents the proportion (percentage) of blue candies in the population.
Next, we'll perform a z-test for proportions since we have a large sample (100 candies) and can assume the sampling distribution to be approximately normal.
The formula for the z-test statistic for proportions is:
z = (p - p) / √[(p * (1 - p)) / n]
where p is the sample proportion, p is the hypothesized proportion, and n is the sample size.
Given that p (sample proportion) is 0.21 (21%), p (hypothesized proportion) is 0.22 (22%), and n (sample size) is 100, let's calculate the z-test statistic:
z = (0.21 - 0.22) / √[(0.22 * (1 - 0.22)) / 100]
z = -0.01 / √[(0.22 * 0.78) / 100]
z ≈ -0.01 / √[0.1716 / 100]
z ≈ -0.01 / √0.001716
z ≈ -0.01 / 0.041442
z ≈ -0.2413 (rounded to four decimal places)
Now, we need to find the critical z-values for a two-tailed test at the 0.01 significance level. Since the significance level is 0.01, we have to split it into two parts (0.005 in each tail). Using a standard normal distribution table or a calculator, we find the critical z-values to be approximately -2.576 and 2.576.
Since our calculated z-test statistic (-0.2413) is not in the rejection region (outside the critical z-values of -2.576 and 2.576), we fail to reject the null hypothesis.
Conclusion:
At the 0.01 significance level, there is not enough evidence to support the candy company's claim that the percentage of blue candies is equal to 22%. The data does not provide sufficient evidence to conclude that the true proportion of blue candies is different from 22%.