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 μ - 2σ or greater than μ + 2σ.

Yes

No

To determine whether it would be significant to get 634 consumers who recognize the Dull Computer Company name in a survey of 800 randomly selected consumers, we need to calculate the mean and standard deviation.

Given that 68% of consumers have heard of the Dull Computer Company, we can calculate the mean:

Mean = 68% of 800 = 0.68 * 800 = 544

To calculate the standard deviation, we use the formula for the standard deviation of a binomial distribution:

σ = √(n * p * (1 - p))

Where:
n = sample size = 800
p = probability of success = 0.68

σ = √(800 * 0.68 * (1 - 0.68))
= √(800 * 0.68 * 0.32)
= √174.08
= 13.18 (approx.)

Now, we can calculate the values that are greater than μ + 2σ and less than μ - 2σ:

μ + 2σ = 544 + 2 * 13.18
= 570.36 (approx.)

μ - 2σ = 544 - 2 * 13.18
= 517.64 (approx.)

Since 634 consumers who recognize the Dull Computer Company name is greater than μ + 2σ (570.36), it would be considered as a significant result according to the criteria mentioned. Therefore, the answer is YES.

To determine if it would be significant to get 634 consumers who recognize the Dull Computer Company name out of a group of 800 randomly selected consumers, we need to calculate the mean and standard deviation of the proportion of consumers who recognize the company.

Given that 68% of consumers have heard of the Dull Computer Company, we can assume that the proportion of consumers who recognize the company follows a normal distribution.

The mean (μ) can be calculated by multiplying the proportion (p) by the sample size (n):
μ = p * n = 0.68 * 800 = 544

The standard deviation (σ) can be calculated using the formula:
σ = sqrt(p * (1 - p) / n)
σ = sqrt(0.68 * (1 - 0.68) / 800) = 0.013

To determine if 634 consumers recognizing the company would be significant, we can calculate the z-score, which tells us how many standard deviations away from the mean the value is:
z = (x - μ) / σ
z = (634 - 544) / 0.013 ≈ 6923.08

Since the calculated z-score is much greater than 2, we can conclude that getting 634 consumers who recognize the Dull Computer Company name out of 800 randomly selected consumers would be significant. The result differs from the mean by more than 2 standard deviations.

No