Hey guys Please help me with this statistics questions It wo

Hey guys. Please help me with this statistics questions. It would be great if someone could show me step by step on how to get to answers. Deeply appreciated. THANKS!

5. You are planning a sample survey of small businesses in your area. You will choose an SRS of businesses listed in the telephone book\'s Yellow Pages. Experience shows that only half the businesses you contact will respond If you contact 150 businesses, it is reasonable to use the B(150,0.5) distribution for the number X who respond. Explain why What is the expected number (the mean) who will reply? What is the probability that 70 or fewer will respond? (Use software or the normal approximation.) How large a sample must you take to increase the mean number of respondents to 100? a. b. c. d.

Solution

A.

Yes, as the number of trials is fixed, the probability of responding is constant (0.5), the number of successes is fixed, and the respondents are independent.

So yes, we can use the binomial distirbution.

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B.

As

mean = n p

Then

mean = 150 * 0.5 = 75 [ANSWER]

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As the standard deviation is

s = sqrt [n p (1 - p)]

Then

s = 6.123724357

Here, as we have \"70 or fewer\", the we get the left tailed area of 70.5.

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    70.5      
u = mean =    75      
          
s = standard deviation =    6.123724357      
          
Thus,          
          
z = (x - u) / s =    -0.734846923      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z >   -0.734846923   ) =    0.231216363 [ANSWER]

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As mean = n p

Then

n = mean / p = 100 / 0.5

= 200 [ANSWER]