The attention span of children (ages 3 to 5) is claimed to be Normally distribut

Question

The attention span of children (ages 3 to 5) is claimed to be Normally distributed with a mean of 15 minutes and a standard deviation of 4 minutes. A test is to be performed to decide if the average attention span of these kids is really this short or if it is longer. You decide to test the hypotheses H0: = 15 versus Ha: > 15 at the 5% significance level. A sample of 10 children will watch a TV show they have never seen before, and the time until they walk away from the show will be recorded. If, in fact, the true mean attention span of these kids is 18 minutes, what is the probability of a Type II error?

Explanation / Answer

Since population sd is known, the test is z-test. The critical value is 1.645.
That is, (Xbar-15)/(4/sqrt10) > 1.645
(Xbar-15) > 2.08
Xbar > 17.08

P(Type II error) = P(Xbar < 17.08 when mean =18)
= P((Xbar-18)/(4/sqrt10) < (17.08 -18)/(4/sqrt10))
= P(Z < -0.73)
= 0.2327

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