Sample standard deviation and population standard deviation

Question

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

Answer 1

(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.