Assume that x

Question

Assume that x has a normal distribution with the specified mean and standard deviation. Find the indicated probability. (Round your answer to four decimal places.)
μ = 5.2; σ = 0.9
P(7 ≤ x ≤ 9) =

 

Im doing the problem as follows:

 

(7-5.2) / .9 = 2         z is greater than or equal to : 2  = .2228

 

(9-5.2) / .9 = 4.2222  z is less than or equal to : 4.2222 = .9999 or 1.0000

 

.2228 - .9999 = .9772    

 

they are saying the problem is incorrect but I have the exact same problem with different numbers but I did it correctly multiple times

 

Can someone tell me what is wrong or what im doing incorrectly

Answer 1

Your value for z=2 of .2228 is really .0228.

 

Your value for z=4.22 should be .00001 because you still want the probability greater than 4.22.

 

Then subtract .0228 - .00001  to get .02279 but rounding gives.0228 back.

 

 

If you use x<4.22 then you also must use also use x<2. P(x<2) = .9772 so this way it is .99999 - .9772 and you still get .0228.

 

Even though in the inequality 7<x<9 you have < and > ;you must get the BOTH probabilities either < or >.

 

 

Please comment back let me know if you understand.

 

 

Answer 2

Normal distributions are symmetrical with a single central peak at the mean (average) of the data. The shape of the curve is described as bell-shaped with the graph falling off evenly on either side of the mean. Fifty percent of the distribution lies to the left of the mean and fifty percent lies to the right of the mean.

The spread of a normal distribution is controlled by the standard deviation. The smaller the standard deviation the more concentrated the data.

 

 

If you add percentages, you will see that approximately:
• 68% of the distribution lies within one standard deviation of the mean (34% above and 34% below the mean).
• 95% of the distribution lies within two standard deviations of the mean (47.5% above and 47.5% below the mean).
• 99.7% of the distribution lies within three standard deviations of the mean (almost 50% above and ~50% below the mean).

 

You are right that 7.5 is 2 standard deviations from the mean of 5.2.  We know that 95% of the data lies within 2 sds--but we look at half of that data falling above the mean, so we divide 95%/2 = 47.5%.  The question asks what is the probability that your data is equal or greater than 2 standard devs however.  So we subtract 50% - 47.5% which is a 2.5% probability.

 

9 on the other hand is more than 4 standard deviations from the mean and we know that 99.7% of the data lies within 3 standard deviations from the mean.  Half of that is almost 50% or 49.85% to be exact.  Again, we are looking for the data past the sds but at that tail end of the curve and so we must subtract 50% - 49.85% = 0.15%

 

Your answer for P(7 ≤ x ≤ 9) = 0.15% < P < 2.5%.....your probability lies between 0.15% and 2.5%.  This would be a bit easier to understand with a graph of a normal curve and distribution in front of you.  Since we are looking at more than 4 sds for the number 9, and data above 3 sds is barely 0.1% on a normal curve graph, you could even say the probability lies between 0.1% and 2.5%.