1

a) a sample of 36 students is selected. What's the probability that the sample mean IQ score of these 36 students is between $95$ and $110$?

b) same as a but what if the sample size is $100$ instead?

My work: X~N(100, 225)

$\sigma = 15$

$\mu = 100$

a) $n = 36$

Using the central limit theorem as $30 \leq n$

$P(95 < \bar X < 110) = P(\cfrac{95-100}{15 / \sqrt {36}} < Z < \cfrac{110-100}{15/\sqrt{36}}) = P(-2 < Z < 4) = .9772 $(approximately using normalcdf on my calculator)

b) $n = 100$

$P(95 < \bar X < 110) = P(\cfrac{95-100}{15 / \sqrt {100}} < Z < \cfrac{110-100}{15/\sqrt{100}}) = P(-3.33 < Z < 6.67) = .9996 $(approximately using normalcdf on my calculator)

Now I think I'm right but these probabilities seem way too high and I wanted to double check with this website. Did I make a mistake somewhere and am I using the right formula? Also my book uses a table but z = 6.67 is not on the table... so I assumed that I had to use my calculator. Is there any other way?

  • Those probabilities do not seem that high to me. $100$ is quite a large sample size. – Toby Mak Oct 11 '20 at 05:31
  • I know that as n increases the distribution becomes more concentrated around $\mu$ so it's natural for the probability to increase if we keep the same range (95 to 110) and raise n to 100. but .9772 for n = 30 and .9996 for n =100 seem a little high. Is this normal? That's almost the entire area! –  Oct 11 '20 at 05:40
  • 1
    @RatChaser92 I mean you already computed the standard deviation of $\bar{X}$ to be $15/\sqrt{36}=2.5$ in (a), and $15/\sqrt{100} = 1.5$ in (b), so $\bar{X}$ is usually quite close to the mean. Nothing alarming in your results. – angryavian Oct 11 '20 at 05:50
  • Ok. Thanks for the reassurance everyone. I guess I just need to get used to stuff like this. –  Oct 11 '20 at 05:53
  • The probability for 100 has to be lower than for 36 - in the 100 you require 100 to concentrate which is a greater demand than the 36. – Moti Oct 11 '20 at 16:25
  • It's lower really? I don't think my math is wrong and my book says that probability should increase as n increases (if within the same range of course). Where did I go wrong then? –  Oct 11 '20 at 18:37

0 Answers0