A survey for brand recognition is done and it is determined that 68% of consumers have heard of Dull Computer Company. If a survey of 800 randomly selected consumers is to be held would it be unusual for 554 of them to recognize Dull Computer Company?

Please show the work so I can understand what formula to use and work it out my self with the other problems.

Well, let's calculate that and see if it's unusual or not. To determine if it's unusual, we can use a statistical test known as a hypothesis test, specifically a test of proportions.

First, let's set up our null and alternative hypotheses:
- Null Hypothesis (H0): The proportion of consumers who recognize Dull Computer Company is equal to 68%.
- Alternative Hypothesis (H1): The proportion of consumers who recognize Dull Computer Company is not equal to 68%.

Next, we need to calculate the standard error of the proportion. The formula for this is:

SE = sqrt[(p * (1 - p)) / n]

Where:
- p is the proportion (68% = 0.68)
- n is the sample size (800)

SE = sqrt[(0.68 * (1 - 0.68)) / 800]
SE = sqrt[(0.2176) / 800]
SE ≈ 0.015

Now, we can calculate the test statistic (z-score) using the observed proportion of 554 out of 800 recognizing Dull Computer Company:

z = (observed proportion - hypothesized proportion) / SE
z = (554/800 - 0.68) / 0.015

z ≈ (0.6925 - 0.68) / 0.015
z ≈ 0.0125 / 0.015
z ≈ 0.833

Looking up the z-score in a standard normal distribution table, we find that the corresponding p-value is approximately 0.2033. This is the probability of observing a test statistic as extreme as z = 0.833, assuming the null hypothesis is true.

Now, if we use a significance level of 0.05 (commonly used in hypothesis testing), we compare the p-value to this level. Since the p-value (0.2033) is greater than the significance level (0.05), we fail to reject the null hypothesis. In other words, there is no strong evidence to suggest that the proportion of consumers recognizing Dull Computer Company is different from 68%.

So, to answer your question, it would not be considered unusual to have 554 out of 800 randomly selected consumers recognize Dull Computer Company, assuming the true proportion of recognition is 68%.

To determine whether it is unusual for 554 out of 800 randomly selected consumers to recognize Dull Computer Company, we can use the binomial probability formula. The formula is:

P(x) = (nCx) * (p^x) * (q^(n-x))

Where:
P(x) is the probability of x successes (recognizing Dull Computer Company)
n is the total number of trials (number of randomly selected consumers)
x is the number of successes (number of consumers recognizing Dull Computer Company)
p is the probability of success in one trial (percentage of consumers who have heard of Dull Computer Company in decimal form)
q is the probability of failure in one trial (1 - p)

In this case, we will use:
n = 800 (total number of randomly selected consumers)
x = 554 (number of consumers recognizing Dull Computer Company)
p = 0.68 (percentage of consumers who have heard of Dull Computer Company in decimal form)
q = 1 - p = 1 - 0.68 = 0.32

Now we can substitute these values into the formula and calculate the probability:

P(554) = (800C554) * (0.68^554) * (0.32^(800-554))

To calculate (800C554), we use the combination formula:
(800C554) = (800! / (554! * (800-554)!))

Calculating this:

(800C554) = (800! / (554! * 246!))
= [(800 * 799 * 798 * ... * 559) / (554! * 246!)]

Now we can substitute all these values into the binomial probability formula:

P(554) = [(800 * 799 * 798 * ... * 559) / (554! * 246!)] * (0.68^554) * (0.32^(800-554))

Calculating this probability will give us an idea of how unusual or likely it is to have 554 consumers recognize Dull Computer Company out of 800 randomly selected consumers.

To determine whether it would be unusual for 554 consumers out of 800 to recognize Dull Computer Company, we can use a hypothesis test.

The first step is to establish the null and alternative hypotheses:

Null Hypothesis (H0): The proportion of consumers recognizing Dull Computer Company is equal to 68% (0.68).
Alternative Hypothesis (Ha): The proportion of consumers recognizing Dull Computer Company is not equal to 68% (0.68).

Next, we calculate the expected proportion of consumers recognizing Dull Computer Company, assuming the null hypothesis is true. This can be done by multiplying the sample size (800) by the null hypothesis proportion (0.68):

Expected Proportion = Sample Size * Null Hypothesis Proportion
Expected Proportion = 800 * 0.68 = 544

Now, we can evaluate whether it would be unusual for 554 consumers to recognize Dull Computer Company. To do this, we can calculate the z-score, which measures how far an observation deviates from the expected proportion in terms of standard deviations:

z = (Observed Proportion - Expected Proportion) / Standard Deviation

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

Standard Deviation = sqrt((Expected Proportion * (1 - Expected Proportion)) / Sample Size)

Plugging in the values:

Standard Deviation = sqrt((544 * (1 - 0.68)) / 800) = sqrt(0.21952) ≈ 0.468

Now, we can calculate the z-score:

z = (0.554 - 0.68) / 0.468 = -0.268

Finally, we can compare the absolute value of the z-score to determine if it is considered unusual.

If the absolute value of the z-score is greater than 1.96 (for a 95% confidence level), then it would be considered unusual. Otherwise, it would not be considered unusual.

In this case, the absolute value of the z-score is |(-0.268)| = 0.268, which is less than 1.96. Therefore, it would not be considered unusual for 554 consumers to recognize Dull Computer Company in a sample of 800 randomly selected consumers.

p = 0.68

n = 800
µ = np
µ = 800(.68) = 544
Standard deviation = √(np(1-p) )
= √(800(.68)(.32) )
= 13.1939
µ±2σ
(544 -2(13.193), 544+2(13.1939))
(517.6, 570.4)
554 is not unusual because it is in interval.