Hello there can please someone help me with this problem Tha

Hello there, can please someone help me with this problem?

Thank you,

8.102 Marketing and Consumer Behavior A successful company usually has high brand-name and logo recognition among consumers. For example, Coca-Cola products are available to 98% of all people in the world, and may have the highest logo recognition of any company. A software firm developing a product would like to estimate the proportion of people who recognize the Linux penguin logo. Of the 952 randomly selected consumers surveyed, 132 could identify the product associated with the penguin. a. Is the distribution of the sample proportion, P, approximately normal? Justify your answer. b. Find a 95% confidence interval for the true proportion of consumers who recognize the Linux penguin. c. The company will market a Linux version of their new software if the true proportion of people who recognize the logo is greater than 0.10. Is there any evidence to suggest the true proportion of people who recognize the logo is greater than 0.10? Justify your answer.

Solution

a)

Yes, because x = 132 > 10, and n - x = 820 > 10.

So yes, it is approximately normal.

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b)

p^ = point estimate of the population proportion = x / n =    0.138655462          
              
Also, we get the standard error of p, sp:              
              
sp = sqrt[p^ (1 - p^) / n] =    0.011200527          
              
Now, for the critical z,              
alpha/2 =   0.025          
Thus, z(alpha/2) =    1.959963985          
Thus,              
Margin of error = z(alpha/2)*sp =    0.02195263          
lower bound = p^ - z(alpha/2) * sp =   0.116702832          
upper bound = p^ + z(alpha/2) * sp =    0.160608092          
              
Thus, the confidence interval is              
              
(   0.116702832   ,   0.160608092   ) [ANSWER]

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c)

YES, because the whole confidence interval is greater than 0.10. [ANSWER]