Question

In: Statistics and Probability

Suppose that a student in this class uses their personalized class data set to find the...

Suppose that a student in this class uses their personalized class data set to find the following confidence interval for the proportion of students in this class who bike more than 5 km to school: (.171, .243). Consider the following statements.

(i) If that same student used the same data set to produce the following confidence interval for the proportion of students who bike more than 5km to school, (.184, .230), then this new confidence interval must have a higher level of confidence than the interval (.171, .243).
(ii) The true proportion of students in this class who bike more than 5km to school might not be in the interval (.171, .243).

(iii) The proportion of students in that same student's personalized class data set who bike more than 5km to school must be in the interval (.171, .243).

Determine which of the above statements are true (1) or false (2).

Solutions

Expert Solution

Solution:

(i) If that same student used the same data set to produce the following confidence interval for the proportion of students who bike more than 5km to school, (.184, .230), then this new confidence interval must have a higher level of confidence than the interval (.171, .243).

Answer: This statement is false because the new confidence interval is (0.184,0.230), which is narrower than the given confidence interval (0.171, 0.243). We know that as the confidence level decreases, the width of the confidence interval narrows down. Therefore, the new confidence interval must have a lower level of confidence than the interval (0.171,0.243). Therefore, the given statement is false.

(ii) The true proportion of students in this class who bike more than 5km to school might not be in the interval (.171, .243).

Answer: This statement is true because the confidence interval is always stated with some level of significance, which is the probability of making the type I error. Therefore, the confidence interval is not necessarily required to hold the true value of the population parameter. Hence the given statement is true.

(iii) The proportion of students in that same student's personalized class data set who bike more than 5km to school must be in the interval (.171, .243)

Answer: The given statement is false because the confidence interval does not give us 100% confidence in seeing the true value of the population parameter in the confidence interval. Hence the given statement is false.


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