A survey for brand recognition is done and it is determined that 68% of consumers have heard of Dull Computer Company. A survey of 800 randomly selected consumers is to be conducted. For such groups of 800,
Would it be significant to get 634 consumers who recognize the Dull Computer Company name? Consider as
Significant any result that differs from the mean by more than 2 standard deviations. That is, significant values
Are either less than $\mu - 2 \sigma$ or greater than $\mu + 2 \sigma$
A) Yes
B) No

Question

Quizplus · Accepted Answer

To determine whether it would be significant to get 634 consumers who recognize the Dull Computer Company name, we need to calculate the mean and standard deviation of the sample proportion.

The mean of the sample proportion is the same as the population proportion, which is 0.68.

The standard deviation of the sample proportion can be calculated using the formula:

$\sigma = \sqrt{\frac{p(1-p)}{n}}$

where p is the population proportion (0.68), and n is the sample size (800).

Plugging in the values, we get:

$\sigma = \sqrt{\frac{0.68(1-0.68)}{800}} = 0.022$

To find the value that differs from the mean by 2 standard deviations, we need to multiply the standard deviation by 2 and add/subtract the result to/from the mean:

$\mu + 2\sigma = 0.68 + 2(0.022) = 0.724$

$\mu - 2\sigma = 0.68 - 2(0.022) = 0.636$

So any result that is greater than 0.724 or less than 0.636 would be considered significant.

To find out if 634 is significant, we need to calculate the sample proportion:

$\hat{p} = \frac{x}{n} = \frac{634}{800} = 0.7925$

This result is greater than the mean (0.68), but is it greater than 2 standard deviations away from the mean?

To find out, we can calculate the z-score:

$z = \frac{\hat{p}-p}{\sigma} = \frac{0.7925-0.68}{0.022} = 5.114$

A z-score of 5.114 is definitely greater than 2, which means that a result of 634 would be considered significant. Therefore, the answer is A) Yes.

A Survey for Brand Recognition Is Done and It Is μ−2σ\mu - 2 \sigmaμ−2σ

A Survey for Brand Recognition Is Done and It Is $\mu - 2 \sigma$