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Question 321:



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You'll need to find two areas under the normal curve given the mean of 250 and standard deviation 10.

  1. Find the normal deviate or z-score for the larger value : (255-250)/10 = a z-score of .5
  2. Find the area under the normal curve using the z-score to percentile calculator, choose 1-sided area. This provides an area of about 69.14%. 
  3. Subtract the smaller area, which following the same steps above is (245-250)/10= -.5 (notice the negative sign) and has area of ~30.85%. Subtracting the smaller gets us 69.14-30.85 = 38.29%

So the probability the sample of 20 is between 245 and 255 is about 38%.  You can also visualize this area by taking a look at the interactive graph of the standard normal curve and enter in the IQ mean of 250 and SD of 10.

One final important qualification to this answer. Your sample size of 20 is somewhat small, meaning the normal distribution might over or under-state the true probability. For smaller samples I recommend using the t-distribution, which is like the normal distribution but takes into account the sample size. If you have a t-table you can look up the areas given the data above.  I ran the computations and found areas of 34.5% and  65.5%. Subtracting these provides a probability of ~31%. So the more conservative answer is 31% and the answer based on the normal distribution is 38%.

See also question 283 and question 318

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