The brand manager for a brand of toothpaste must plan a campaign designed to increase brand recognition. He wants to first determine the per

Question

The brand manager for a brand of toothpaste must plan a campaign designed to increase brand recognition. He wants to first determine the percentage of adults who have heard of the brand. How many adults must he survey in order to be 95​% confident that his estimate is within five percentage points of the true population​ percentage?

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Sophia 43 mins 2022-01-07T01:18:05+00:00 1 Answer 0 views 0

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    2022-01-07T01:19:37+00:00

    Answer:

    n=\frac{0.5(1-0.5)}{(\frac{0.05}{1.96})^2}=384.16  

    And rounded up we have that n=385

    Step-by-step explanation:

    Previous concepts

    A confidence interval is “a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval”.  

    The margin of error is the range of values below and above the sample statistic in a confidence interval.  

    Normal distribution, is a “probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean”.  

    The population proportion have the following distribution

    p \sim N(p,\sqrt{\frac{p(1-p)}{n}})

    Solution to the problem

    In order to find the critical value we need to take in count that we are finding the interval for a proportion, so on this case we need to use the z distribution. Since our interval is at 95% of confidence, our significance level would be given by \alpha=1-0.95=0.05 and \alpha/2 =0.025. And the critical value would be given by:

    z_{\alpha/2}=-1.96, z_{1-\alpha/2}=1.96

    The margin of error for the proportion interval is given by this formula:  

     ME=z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}    (a)  

    And on this case we have that ME =\pm 0.05 (or within 5% points of the true population percentage)  and we are interested in order to find the value of n, if we solve n from equation (a) we got:  

    n=\frac{\hat p (1-\hat p)}{(\frac{ME}{z})^2}   (b)  

    For this case since we don’t have prior distribution, we can use an estimator for  \hat p =0.5. And replacing into equation (b) the values from part a we got:

    n=\frac{0.5(1-0.5)}{(\frac{0.05}{1.96})^2}=384.16  

    And rounded up we have that n=385

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