# Normal distribution and the probability of type I errors

#1
New pupils entering a large secondary school take a general knowledge test during their first week. The mean score achieved on this test is 46.7 with a standard deviation of 14.3. At the beginning of the second term, pupils were asked if they would like to be considered for a team to represent the school in a general knowledge quiz. The test scores of a random sample of the pupils who did wish to be considered for the quiz team were:
38, 63, 79, 91, 42, 53, 84

a) Test, using the 5% significance level, whether the mean test score of those pupils who wished to be considered for the quiz team exceeded 46.7. Assume that the sample of scores is from a normal distribution with standard deviation 14.3. Interpret your conclusion in context:

From sample, sample mean = 64.3

The sample mean is normally distributed by N(46.7, 14.3^2/7)

H0: μ = 46.7

H1: μ > 46.7

Φ^-1 (0.95) = 1.645 = z0.95 where Z is normally distributed by N(0,1)

z = (64.3 - 46.7)/(14.3/root(7)) = 3.256

3.256 > 1.645

Therefore, reject H0.

There is sufficient evidence at 5% level of significance that the mean test score has exceeded 46.7

b) A further random sample of those pupils who wished to be considered for the quiz team is to be taken and the test in part (a) repeated. State, with an explaination, the probability of making a Type I error in this test if the mean test score for all pupils who wished to be considered for the quiz team is:

i) 46.7

ii) 56.7

How do I do parts b) i) and ii)?
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