One of the ways Americans relieve stress is to reward themselves with sweets. Su

Question

One of the ways Americans relieve stress is to reward themselves with sweets. Suppose a study claims 52% of Americans admit to overeating sweets when stressed. Suppose also that the 52% figure is correct for the population and that return samples of size n=100 Americans are selected.

A). Does the distribution of ^p have an approximately normal distribution? If so, what are its mean and standard deviation?

B) Using the normal approximation of ^p without the continuity correction, what's the probability of getting a sample (n=100) with ^p greater than .6?

C) Using the normal approximation of the binomial distribution with the continuity correction, what's the probability of getting a sample (n=100) with ^p greater than .6?

D) Using the exact binomial calculation, what's the probability of getting sample (n=100) with ^p greater than .6?

Explanation / Answer

a)

Yes, because np and n(1-p) are both greater than 10.

u = mean = p =    0.52   [ANSWER]  
          
s = standard deviation = sqrt(p(1-p)/n) =    0.049959984   [ANSWER]  

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b)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    0.6      
u = mean = p =    0.52      
          
s = standard deviation = sqrt(p(1-p)/n) =    0.049959984      
          
Thus,          
          
z = (x - u) / s =    1.601281538      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   1.601281538   ) =    0.054657288 [ANSWER]

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c)

We first get the z score for the critical value, which is 60 + 0.5:          
          
x = critical value =    60.5      
u = mean = np =    52      
          
s = standard deviation = sqrt(np(1-p)) =    4.995998399      
          
Thus, the corresponding z score is          
          
z = (x-u)/s =    1.701361634      
          
Thus, the left tailed area is          
          
P(z <   1.701361634   ) =    0.04443755 [ANSWER]

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d)

Note that P(more than x) = 1 - P(at most x).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    100      
p = the probability of a success =    0.52      
x = our critical value of successes =    60      
          
Then the cumulative probability of P(at most x) from a table/technology is          
          
P(at most   60   ) =    0.956056455
          
Thus, the probability of at least   61   successes is  
          
P(more than   60   ) =    0.043943545 [ANSWER]

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