Question

1) Assume that a sample is used to estimate a population mean μμ. Find the 98% confidence interval for a sample of size 514 with a mean of 70.8 and a standard deviation of 14.5. Enter your answer accurate to one decimal place (because the sample statistics are reported accurate to one decimal place).

__ < μ < __

2) Assume that a sample is used to estimate a population mean μμ. Find the 99.5% confidence interval for a sample of size 457 with a mean of 60.2 and a standard deviation of 15.6. Enter your answer accurate to one decimal place (because the sample statistics are reported accurate to one decimal place).

__ < μ < __

3) If n=16, ¯x(x-bar)=30, and s=3, find the margin of error at a 95% confidence level.
Give your answer to two decimal places.

Answer 1

Solution :

1) Given that,

Point estimate = sample mean = = 70.8

sample standard deviation = s = 15.6

sample size = n = 514

Degrees of freedom = df = n - 1 = 514 - 1 = 513

At 98% confidence level

= 1 - 98%

=1 - 0.98 =0.02

/2 = 0.01

t/2,df = t_0.01,513 = 2.334

Margin of error = E = t_/2,df * (s /n)

= 2.334 * ( 15.6 / 514)

Margin of error = E = 1.6

The 98% confidence interval estimate of the population mean is,

- E < < + E

70.8 - 1.6 < < 70.8 + 1.6

( 69.2 < < 72.4 )

2) Given that,

Point estimate = sample mean = = 60.2

sample standard deviation = s = 15.6

sample size = n = 457

Degrees of freedom = df = n - 1 = 457 - 1 = 456

At 99.5% confidence level

= 1 - 99.5%

=1 - 0.995 =0.005

/2 = 0.0025

t/2,df = t_0.0025,456 = 2.821

Margin of error = E = t_/2,df * (s /n)

= 2.821 * (15.6 / 457)

Margin of error = E = 2.1

The 99.5% confidence interval estimate of the population mean is,

- E < < + E

60.2 - 2.1 < < 60.2 + 2.1

( 58.1 < < 62.3 )

3) Given that,

Point estimate = sample mean = = 30

sample standard deviation = s = 3

sample size = n = 16

Degrees of freedom = df = n - 1 = 16 - 1 = 15

At 95% confidence level

= 1 - 95%

=1 - 0.95 =0.05

/2 = 0.025

t/2,df = t_0.025,15 = 2.131

Margin of error = E = t_/2,df * (s /n)

= 2.131 * (3 / 16)

Margin of error = E = 1.60