Question

About 20% of US adults don't get enough sleep. Samples of 150 adults are taken.

a) Show the sampling distribution of p. (shape, mean, standard error)
b) What is the probability that the sample portion will be witin +/- .03 of the population proportion?

c) What happens to the probability as the precision level is increased ( from 3% to 5%)?

Answer 1

p = proportion of US adults don't get enough sleep = 0.20

n = sample size = 150

(a) Sampling distribution of p

Here sample size is large so according to the central limit theorem, the sampling distribution of sample proportion phat is normally distributed with mean and standard deviation

Shape of distribution: Normal

Mean:

Standard error:

b) The probability that the sample portion will be within +/- .03 of the population proportion that is p - 0.03 < phat < p + 0.03

The z scores for both the lower limit is,

The z score for the upper limit is,

Probability becomes P(-0.92 < z < 0.92)

Using the z table the probability for z score -0.92 is 0.1788 and the probability for z score 0.92 is 0.8212

The required probability is 0.8212 - 0.1788 = 0.6424

Therefore, the probability that the sample portion will be within +/- .03 of the population proportion is 0.6424

c) As the precision level increases the probability also increases since the distance from the mean is increases.

That is P( p - 0.05< phat < p + 0.05)

The z scores for the limits are

The probabilities using z score table for z score -1.53 and 1.53 are 0.0630 and 0.9370

the required probability is 0.9370 - 0.0630 = 0.874

That is the probability increases and the precision level increased from 3% to 5%