Andy is a restaurant owner: He believes 85% of his customers are satisfied…

Question

Andy is a restaurant owner: He believes 85% of his customers are satisfied with the food quality of his restaurant: From a random sample of 96 customers, what are the following probabilities? (Round to 2 decimal places for all z-values and round all other answers to 4 decimal places, if needed.)

a) What is the probability that less than 82 customers are satisfied with the food quality?

b)What is the probability that at least 82 customers are satisfied with the food quality?

c) What is the probability that the sample proportion of customers who are satisfied with the food quality is between 82% and 90%?

Answer 1

a) P(p hat < 0.854)= 0.5438

 

b) P(p hat > 0.854)= 0.4562

 c) P(0.82 < p hat < 0.9) = 0.7086



Step-by-step explanation

Values are given as,

n= 96

po= 0.85

σp​=((1−po​)∗po​)/n​ =((1−0.85)∗0.85)/96​ = [0.0364]

 

 

 

a) p hat= (82/96)= 0.854

 

We need to find P(p hat < 0.854)

Using below formula 

P[z<(p^​−po​)/σp​]

Put the values in the formula,

P[(z < (0.854 - 0.85)/0.0364]

= (z < 0.11)

z score is 0.11

Using z table, we get,

P(z < 0.11) = 0.5438

Hence, P(z < 0.11) = 0.5438

 

 

 

b) p hat= (82/96)= 0.854

We need to find P(p hat > 0.854)

Using below formula 

P[z<(p^​−po​)/σp​]

Put the values in the formula,

P[(z < (0.854 - 0.85)/0.0364]

= (z < 0.11)

z score is 0.11

Using z table, we get,

P(z < 0.11) = 0.5438

Now P(z > 0.11) = 1 - P(z < 0.11)

Now P(z > 0.11) = 0.4562

Hence, P(z > 0.11) = 0.4562

 

 

 

 

c) 

We need to find P(0.82 < p hat < 0.9)

Using below formula 

P[(p^​1​−po​)/σp​<z<(p^​2​−po​)/σp​]

Put the values in the formula,

P[(0.82 - 0.85)/0.0364 < z < (0.9 - 0.85)/0.0364]

= (-0.82 < z < 1.37)

z scores are (-0.82, 1.37)

Using below formula,

P(a < z < b) = P(z < b) - P(z < a)

Put the values in above formula,

P(-0.82 < z < 1.37) = P(z < 1.37) - P(z < -0.82)

Using z table, we get,

P(z < 1.37) = 0.91466

P(z < -0.82) = 0.20611

So, P(-0.82 < z < 1.37) = 0.91466 - 0.20611

= 0.7086

Hence, P(-0.82 < z < 1.37) = 0.7086