Statistise on Confidence intervals and Hypothesis testing

Question

To study the average rating of chess players at a tournament, we have a random sample of 360 players. The sample mean is 1660 and the true standard deviation is σ = 360. a. Construct an approximate 95% confidence interval for the average rating of the tournament’s chess players. b. Construct an approximate 99% confidence interval for the average rating of the tournament’s chess players. c. Perform a hypothesis test for H0 : µ = 1650 vs. HA : µ= 1650 at a 1% level. Do this using the three approaches we learned (p-value, critical value, inverting a confidence interval). d. Perform a hypothesis test for H0 : µ ≥ 1650 vs. HA : µ < 1650 at a 5% level

Answer 1

a) To construct a confidence interval use the following formula

Confidence Interval = mean ± (Z score)(std dev)/(√sample size)

For a 95% CI we are 2 std deviations each direction from the mean. So we get the following

CI = 1660 ± 2(360/√360) Which is 1660 ± 37.95

So a 95% CI would be 1622.05 ≤ μ ≤ 1697.95

To interpret this, 95% of all players at the tournament fall in these ratings.

b) For a 99% CI its the same idea except the Z score is 2.576.

CI = 1660 ± 2.576(360/√360) which is 1660 ± 48.88 I'll leave you to find the CI.

c) I'm assuming that HA : μ≠1650 is what you meant.

If we are interested in a 1% level and we have HA as such then this is a two-tailed test since values greater than and less than our interval violate the null hypothesis. This means we are interested in seeing the z-score which serves as a critical value with 0.5% left on each size of the bell curve. This means our critical values are z=±2.807.

Calculate the z score of a 1650 rating and then compare it to the critical value. If the z-score it outside of our interval we reject the null hypothesis.

d) Since the null says that μ≥1650 we have a one tailed test on the left hand side. Using the same logic as c find the critical value z-score for there to be 5% on the left of the curve. Then compare the z-score of 1650 to the critical value you find.

Hope this helps!