Question

Can someone please show me step by step on how to solve this? My answer that I picked is wrong.

Answer 1

31 is two standard deviations below the mean, since

31 = 35 - 4 = 35 - 2×2

33 is one standard deviation below the mean, since

33 = 35 - 2

So, we have

`P(31 < X < 33) = P(-2 < Z < -1)`

where
`X` has the given normal distribution (mean 35, s.d. 2) and
`Z` has the standard normal distribution (mean 0, s.d. 1).

Recall the empirical (68-95-99.7) rule, which says

• 
  `P(-1 < Z < 1) \approx 68\%`
• 
  `P(-2 < Z < 2) \approx 95\%`
• 
  `P(-3 < Z < 3) \approx 99.7\%`

The normal distribution is also symmetric about its mean, so

• 
  `P(0 < Z < 1) = P(-1 < Z < 0) \approx 34\%`
• 
  `P(0 < Z < 2) = P(-2 < Z < 0) \approx 47.5\%`
• 
  `P(0 < Z < 3) = P(-3 < Z < 0) \approx 49.85\%`

Now,

`P(-2 < Z < -1) = P(-2 < Z < 0) - P(-1 < Z < 0) \approx 47.5\% - 34\% = 13.5\%`

so the 351 values between 31 and 33 make up about 13.5% of the total body of data in the distribution. Then there must be

`(351)/(13.5\%) = (351)/(0.135) = \boxed{2600}`

values in the whole data set. (A)