In December 2004, 48% of students in high school were satisfied with the lunches
ID: 3048991 • Letter: I
Question
In December 2004, 48% of students in high school were satisfied with the lunches supplied through the school. In May 2010, an organization conducted a poll of 792 students in high school and asked if they were satisfied with the lunches supplied through the school. Of the 792 surveyed, 341 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 alphaequals=0.10 level of significance, compute the probability of making a Type II error, beta, if the true population proportion is 0.43. What is the power of the test? (c) Redo part (b) if the true population proportion is 0.47.
Explanation / Answer
Below are the null and alternate hypothesis
H0: p >= 0.48
H1: p < 0.48
pcap = 341/792 = 0.4306
Test statistics,
z = (0.4306 - 0.48)/sqrt(0.48*0.52/792)) = -2.7827
p-value = 0.0027 (uisng standard z table)
As p-value is less than the significance level of 0.1, we reject the null hypothesis.
This means there are significant evidence to conclude that the proportion of students satisfied with the quality of lunches has decreased.
a)
Type II error will occur in this test if we fail to reject the null hypothesis though the true proportion is less than 0.48.
b)
Hypothesis test is being done above.
c)
p0 (hypothesised proportion) 0.48 SE = sqrt(p*(1-p)/n) 0.01775251 n 792 alpha 0.1 sample/true proportion 0.43 Std. Error. SE 0.0178 Zcritical -1.282 Xcritical 0.46 Beta or type II error is the probability of fail to reject the null hypothesis P(p>0.46) 0.0624 Hence type II error probability is 0.0624 Power of the test 0.9376