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Suppose that in a random selection of 100 colored​ candies, ​21% of them are blue. The candy company claims that the percentage of blue candies is equal to ​22%. Use a 0.01 significance level to test that claim.

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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%.

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