Question

2. From a sample of size 250, number of females is 173. What is the point estimate of population proportion? Provide an answer with 3 decimal points.

3. Which one is the point estimate of population mean? a. sample mean b. sample proportion c. sample median d. sample maximum

4. A survey of 1,206 people asked: “What would you do with an unexpected tax refund?” Forty-seven percent responded that they would pay off debts (Vanity Fair, June 2010). if the population size is 5000, then write down the upper bound of 95% confidence interval of the population proportion.

5. An article in the National Geographic News (“U.S. Racking Up Huge Sleep Debt,” February 24, 2005) argues that Americans are increasingly skimping on their sleep. A researcher in a small Midwestern town wants to estimate the mean weekday sleep time of its adult residents. He takes a random sample of 80 adult residents and records their weekday mean sleep time as 6.4 hours. Assume that the population standard deviation is fairly stable at 1.8 hours. Write down the upper bound of 95% confidence interval.

Answer 1

2. Given a sample of size n = 250, the number of females is X = 173 hence the point estimate of the population proportion is the sample proportion which is calculated as:

3.  The point estimate of a population mean is the sample mean of the sample taken from the population.

4. Given that out of N =1,206 people asked Forty-seven percent (47%) of responded that they would pay off debts (Vanity Fair, June 2010) hence the sample proportion is = 0.47.

Now the confidence interval is calculated as:

The Zc at 95% confidence level is calculated using the excel formula for normal distribution which is =NORM.S.INV(0.975), Thus the Zc is computed as 1.96

Thus the confidence interval is calculated as:

Thus the upper bound of 95% confidence interval of the population proportion is 0.498

5) An article in the National Geographic News (“The U.S. Racking Up Huge Sleep Debt,” February 24, 2005) argues that Americans are increasingly skimping on their sleep.

A researcher takes a random sample of n = 80 adult residents and records their weekday mean sleep time as M = 6.4 hours. Assume that the population standard deviation is fairly stable at s = 1.8 hours.

Now the confidence interval is calculated as:

μ = M ± Z(s_M)

The Z score is applicable since the sample size is greater than 30 and the population standard deviation is known.

where:

M = sample mean
Z = Z statistic determined by the confidence level
s_M = standard error = √(s²/n)

Thus

M = 6.4
Z = 1.96
s_M = √(1.8²/80) = 0.2

μ = M ± Z(s_M)
μ = 6.4 ± 1.96*0.2
μ = 6.4 ± 0.394

\[6.006, 6.794

Thus the upper bound of 95% confidence interval is 6.794.