In: Statistics and Probability
In December 2004, 54% of students in high school were satisfied with the lunches supplied through the school. In May 2010, an organization conducted a poll of 1045 students in high school and asked if they were satisfied with the lunches supplied through the school. Of the 1045 surveyed, 502 indicated they were satisfied. Does this suggest the proportion of students satisfied with the quality of lunches has decreased?
(a) What does it mean to make a Type II error for this test?
(b) If the researcher decides to test this hypothesis at the alpha equals 0.10 level of significance, compute the probability of making a Type II error, beta, if the true population proportion is 0.48. What is the power of the test?
(c) Redo part (b) if the true population proportion is 0.52.
Hypothesis Test:
= p = 0.54
=
=
= 0.0154
p-value = 0.0001 < 0.05 i.e. we can reject H0 and hence we can say that the proportion of students satisfied with the quality of lunches has decreased significantly.
A.
Type II error: = Failure to reject a false null hypothesis.
Thus, in this case, it would mean that we concluded that the proportion of students satisfied with the quality of lunches has not decreased significantly when actually it has.
B.
Critical values for this test:
( - 0.54)/0.01542 = -1.28 (At 0.1 level of significance)
So,
= 0.54 - 1.28*0.01542
= 0.5203
Type II error, = P(>0.5203|p=0.48)
= P(Z>(0.5203-0.48)/0.015455)
= P(Z>2.6076)
= 0.0046
Power of the test = 1-
= 1 - 0.0046
= 0.9954
C.
Type II error, = P(>0.5203|p=0.52)
= P(Z>(0.5203-0.52)/0.015455)
= P(Z>0.0194)
= 0.4923
Power of the test = 1-
= 1 - 0.4923
= 0.5077
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