Question

Students should select one problem and solve. Post should include a detailed solution and explanation.

1. A random sample of 56 fluorescent light bulbs has a mean life of 645 hours. Assume the population standard deviation is 31 hours. Construct a 95% confidence interval for the population mean.

2. The standard IQ test has a mean of 96 and a standard deviation of 14. We want to be 99% certain that we are within 4 IQ points of the true mean. Determine the required sample size.

3. A random sample of 10 parking meters in a beach community showed the following incomes for a day. Assume the incomes are normally distributed.

$3.60   $4.50   $2.80   $6.30   $2.60   $5.20   $6.75   $4.25   $8.00   $3.00

Find a 90% confidence interval for the incomes of all parking meters in the beach community.

4. The numbers of advertisements seen or heard in one week for 30 randomly selected people in the United States are listed below. Construct a 95% confidence interval for the true mean number of advertisements. Assume that σ = 159.5

598      494      441      595      728      690      684      486      735      808

481      298      135      846      764      317      649      732      582      677

734      588      590      540      673      727      545      486      702      703

5. The number of wins in a season for 32 randomly selected professional football teams are listed below. Construct a 90% confidence interval for the true mean number of wins in a season.

9          9          9          8          10        9          7          2         

11        10        6          4          11        9          8          8         

12        10        7          5          12        6          4          3         

12        9          9          7          10        7          7          5   

Answer 1

Solution(1)
No. of sample =56
Mean =645
SD = 31

So 95% confidence interval can be calculated as
Mean +/- Zalpha/2 *SD/sqrt(n)
Given alpha=0.05 and alpha/2 =0.025 and Z0.025 =1.96
645 +/- 1.96*(31)/sqrt(56)
645 +/- 1.96*31/7.4833
645 +/- 8.1194125

So 95% confidence interval is 636.88 to 653.12

Solution(2)
alpha=0.01 and alpha/2 =0.005 so Zalpha/2= 2.575
Standard deviation = 14
Confidence interval = (96-4) to (96+4) so it is (92 to 100) and Margin of error = 4
So margin of error can be calculated as
4 = 2.575*14sqrt(n)
n = (2.575*14/4)^2 = 9.0125 *9.0125 = 81.225 or 81
So sample size is 81.