Question

There have been many studies that suggest global climate change is affecting the population of polar bears. One measure of the health of a polar bear population is weight, and the mean weight for a male polar bear is approximately 900 pounds with population deviation 78 pounds. A recent study suggests that the polar bears in the Chukchi Sea are doing very well despite changes in the climate. A random sample of 14 male polar bears from the Chukchi Sea was obtained. The sample mean weight was 1016 pounds. Assume the underlying weight distribution of polar bears is normal. a Suppose we wish to perform a significance test with Ha : µ > 900, what is your decision at the significance level of 0.01? b. What is your decision at the significance levle of 0.05 and 0.001 respectively? d. In part a, what is the type I error rate α? e. In part a, if the true population mean is 1000 pounds, what is the type II error rate β?

Answer 1

a) The test statistic z = ()/()

= (1016 - 900)/(78/)

= 5.56

P-value = P(Z > 5.56)

= 1 - P(Z < 5.56)

= 1 - 1 = 0

Since the p-value is less than the significance level (0 < 0.01),  

So we should reject the null hypothesis.

B) At 0.05 significance level, since the p-value is less than the significance level (0 < 0.05), so we should reject the null hypothesis.

At alpha = 0.001, since the p-value is less than the significance level (0 < 0.001), so we should reject the null hypothesis.

D) alpha = 0.01

E) At alpha = 0.01, the critical value is z_0.99 = 2.33

Z_crit = 2.33

Or, ( - )/() = 2.33

Or, ( - 900)/(78/) = 2.33

Or, = 2.33 * (78/) + 900

Or, = 948.572

= P( < 948.572)

= P(( - )/() < (948.572 - )/())

= P(Z < (948.572 - 1000)/(78/))

= P(Z < -2. 47)

= 0.0068