In August 2002, 47% of parents who had children in grades K-12 were satisfied wi
ID: 3394298 • Letter: I
Question
In August 2002, 47% of parents who had children in grades K-12 were satisfied with the quality of education the students receive. In September 2014 the Gallup organisation conducted a poll of 1013 parents who have children in grades K-12 and asked if they were satisfied with the quality of education the students receive. Of the 1013 surveyed 437 indicated they were satisfied. Is there evidence at the alpha = 0.05 significance level that parents' feelings have changed? Based on the sample, construct a 95% confidence interval for the percent of who are satisfied. Based on this interval, is there evidence parents feelings have changed?Explanation / Answer
a)
Formulating the null and alternatuve hypotheses,
Ho: p = 0.47
Ha: p =/= 0.47
As we see, the hypothesized po = 0.47
Getting the point estimate of p, p^,
p^ = x / n = 0.431391905
Getting the standard error of p^, sp,
sp = sqrt[po (1 - po)/n] = 0.015681303
Getting the z statistic,
z = (p^ - po)/sp = -2.462046378
As this is a 2 tailed test, then, getting the p value,
p = 0.01381468
significance level = 0.05
As P < 0.05, we REJECT THE NULL HYPOTHESIS.
Thus, there is significant evidence that the proportion of parents satisfied with the quality of education is now different from 47%, at 0.05 level. [CONCLUSION]
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b)
Note that
p^ = point estimate of the population proportion = x / n = 0.431391905
Also, we get the standard error of p, sp:
sp = sqrt[p^ (1 - p^) / n] = 0.01556101
Now, for the critical z,
alpha/2 = 0.025
Thus, z(alpha/2) = 1.959963985
Thus,
Margin of error = z(alpha/2)*sp = 0.03049902
lower bound = p^ - z(alpha/2) * sp = 0.400892885
upper bound = p^ + z(alpha/2) * sp = 0.461890925
Thus, the confidence interval is
( 0.400892885 , 0.461890925 )
As 0.47 is not inside this interval, yes, there is evidence that parents feelings have changed. [CONCLUSION]