Question

2. According to a recent study, 40% of Americans believe that marriage is necessary for a happy life.

1. (a) Suppose a random sample of 500 Americans is asked about their opinion on mar- riage. What is the sampling distribution of pˆ, the proportion who believe marriage is necessary for a happy life? Verify all necessary assumptions.

2. (b) What is the probability that in the random sample of 500 Americans, between 40% and 43% believe marriage is necessary for a happy life?

3. (c) Suppose you did not know the true population proportion of Americans that believe marriage is necessary for a happy life, and so you collect a sample to estimate it. In your sample of 500 Americans, 230 believe marriage is necessary for a happy life. Based on this, compute a 90% confidence interval for the population proportion. Interpret the confidence interval.

4. (d) If you increased the sample size, what would happen to the width of your confi- dence interval in part c)?

5. (e) If you increased the confidence level, what would happen to the width of your confidence interval in part c)?

Answer 1

a) The values of np and nq are computed here as:
np = 500*0.4 = 200 > 5 and n(1-p) = n - np = 300 > 5
Therefore the normality assumptions are satisfied here, and therefore the sampling distribution of sample proportion here is given as:

This is is the required sampling distribution of proportion here.

b) The probability that between 40% and 43% believe marriage is necessary for a happy life is computed here as:

P( 0.4 < p < 0.43)

Converting it to a standard normal variable, we have here:

Getting it from the standard normal tables, we have here:

Therefore 0.4145 is the required probability here.

c) The sample proportion here is computed as:
P = x/n = 230/500 = 0.46

Also for 90% confidence level, we have from the standard normal tables:
P( -1.645 < Z < 1.645) = 0.9

Therefore the confidence interval here is obtained as:

d) As the sample size increases, the margin of error decreases, therefore the confidence interval width also decreases. Therefore the width of the confidence interval would decrease here.

e) As the confidence level increases, the critical z value increases and hence the margin of error increases and therefore the width also increases.