Question

A random sample of n = 300 observations from a binomial population produced x = 223 successes. Find a 90% confidence interval for p. (Round your answers to three decimal places.)

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Interpret the interval.

In repeated sampling, 10% of all intervals constructed in this manner will enclose the population proportion.There is a 90% chance that an individual sample proportion will fall within the interval.    In repeated sampling, 90% of all intervals constructed in this manner will enclose the population proportion.There is a 10% chance that an individual sample proportion will fall within the interval.90% of all values will fall within the interval.

You may need to use the appropriate appendix table or technology to answer this question

Answer 1

pˆ = 263/300 = .8767 (We use an estimate for p since we don't know the true population value)

SE = (pq/n) = (.8767)(1?.8767)/300 = .019

Here = 0.1

Hence Z score = 1.645.

pˆ + z_{/2 *} SE = .8767+- (1.645)(.019) = [0.845, 0.908]

This is the 90% CI.

Interpretation:

It means that there is a 90% probability/we can say with 90% confidence that pˆ is within [0.845, 0.908]

i.e. When we repeat this method of finding interval over and over again, it will produce intervals which will include the true proportion 'p', 90% of the time.

i.e. In repeated sampling, 90% of all intervals constructed in this manner will enclose the population proportion.