4 It is commonly believed that for healthy adult males the h

[4] It is commonly believed that for healthy adult males, the heart rate is normally distributed with an average of 80 beats per minute, with a standard deviation of 3 beats. If a subject is selected at random, find the probability that ... (a) the person has a heart rate between 80 and 87 beats per minute (b) the person has a heart rate of at most 77 beats per minute (c) State the central limit theorem (d) If we randomly sample 10 healthy adult males, what is the probability that the sample mean is less that 75 beats per minute.

Solution

A.

We first get the z score for the two values. As z = (x - u) sqrt(n) / s, then as          
x1 = lower bound =    80      
x2 = upper bound =    87      
u = mean =    80      
n = sample size =    1      
s = standard deviation =    3      
          
Thus, the two z scores are          
          
z1 = lower z score =    0      
z2 = upper z score =    2.333333333      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.5      
P(z < z2) =    0.990184671      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.490184671   [ANSWER]

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B.

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    77      
u = mean =    80      
n = sample size =    1      
s = standard deviation =    3      
          
Thus,          
          
z =    -1      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z >   -1   ) =    0.158655254 [ANSWER]

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C.

As the number of samples increase, the mean of these samples will be the population mean, and they will be normally distributed.

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D.

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    75      
u = mean =    80      
n = sample size =    10      
s = standard deviation =    3      
          
Thus,          
          
z =    -5.270462767      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z >   -5.270462767   ) =    6.80401E-08 or 0.00000006804 [ANSWER]