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Hypothesis Testing, Confidence Intervals and Levels of Significance

In a statistics lecture, students are asked whether or not they enjoyed doing statistics. Random sample of 50 students was taken and 30 of them said that they enjoyed doing statistics. The lecturer claimed that more than 50% of the students enjoyed doing statistics.

(i) Test, at the 5% level of significance, whether or not the lecturer's claim is reasonable. State the null and alternative hypotheses of the test.

(ii) Find a 95% and a 99% confidence intervals for the true proportion of students who enjoyed doing statistics. Which one is wider? Why?

(iii) If the lecturer would like the difference between the true proportion and the sample
proportion to be less than 5%, what is the minimum number of students he should ask Use alpha = 0.05

Solution Preview

First, we need to find the sample mean of the proportion of students who enjoyed doing statistics.
p-hat=x/n=30/50=0.60
Note. The sample size n=50.

(i) Hypotheses.

Null hypothesis H0: p=0.50
Alternative hypothesis Ha: p>0.50

Use z-test. We can calculate the z-statistic
z=(p-hat-0.50)/sqrt[0.50*(1-0.50)/n]
=(0.60-0.50)/sqrt[0.5*0.5/50]=1.4142

At the 5% level of significance, the critical value ...

Solution Summary

Hypothesis testing, confidence intervals and levels of significance are investigated.

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