For males the exprected pulse rate is 70/sec and the standard deviation is 10/se

Question

For males the exprected pulse rate is 70/sec and the standard deviation is 10/sec. For women the expected pulse rate is 77/sec and sd is 12/sec.

Let X = sample average pulse rate for a random sample of 40 men and let Y = the sample average pulse rate for a random sample of 36 women.

a) What is the approximate distribution of X? of Y?

b) What is the approximate distribution of X-Y?

c) Calculate (approximately) the probability P(-2 <= X-Y <= 1)

d) Calculate (approximately) the probability P(X-Y <= -15). If you actually observed X-Y<=-15, would you doubte that mu1 - mu2 = -7.

Explanation / Answer

a)

Let N(u, sigma) be a normal distribution of mean u and standard deviation sigma.

Hence, for X, by central limit theorem:

X~N(70, 10/sqrt(40))

or

X~N(70,1.58113883) [ANSWER]

For Y:
  
Y~N(77, 12/sqrt(36))

or

Y~N(77,2) [ANSWER]

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b)

Hence,

X-Y~N(70-77, sqrt(1.5811^2+2^2))

or

X-Y~N(-7, 2.549509757) [ANSWER]

****************

c)

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    -2      
x2 = upper bound =    1      
u = mean =    -7      
          
s = standard deviation =    2.549509757      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    1.961161351      
z2 = upper z score = (x2 - u) / s =    3.137858162      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.975069898      
P(z < z2) =    0.999149064      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.024079166   [ANSWER]
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d)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    -15      
u = mean =    -7      
          
s = standard deviation =    2.549509757      
          
Thus,          
          
z = (x - u) / s =    -3.137858162      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   -3.137858162   ) =    0.000850936 [ANSWER]

YES, I WILL DOUBT IT, AS THIS IS A VERY SMALL PROBABILITY. [ANSWER]
  

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