3
$\begingroup$

The average temperature of a particular tropical island is normally distributed with a mean of 74 degrees and a variance of 9 degrees

(a) If a random sample of 16 days has been taken, what is the probability that at least 12 of them will have temperatures higher than 70 degrees?

(b) If a random sample of 16 days has been taken, what is the probability that the sample standard deviation is within 1 unit from the population standard deviation?

for part (a) i get 0.99.

part b is not something with which i feel very acquainted. could anyone tell me the approach?

I feel the problem is asking

$$ P(-\sigma < s < \sigma ) $$ And I can use the fact that

$$ \frac{(n-1)s^2}{\sigma^2} \ follows \ X^2(n-1) $$

But I am not sure how to proceed, because I do not know what is s

$\endgroup$
0
$\begingroup$

(a) Let $p = P\{X_i > 70\} \approx .91$ (It has to be pretty large because 70 is more than a standard deviation below the mean.) Then the answer to the question is obtained from a binomial distribution with this $p$ and $n = 16$. Your answer seems about right.

(b) Bounds on the sample standard deviation using the chi-squared distribution. First, to be fussy, the units of the variance are squared degrees. We know $\sigma = 3$ and $n = 16.$

I believe the problem is asking for $$P\{2 < S < 4\} = p\{4 < S^2 < 16\} =\cdots = P\{20/3 < (n-2)S^2/\sigma^2 < 80/3\}.$$

You should be able to fill in the missing steps and evaluate this using software. I got about .93 using the chi-squared distribution with df = 15.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.