Question

In: Statistics and Probability

A simple random sample of size n is drawn. The sample​ mean, x overbarx​, is found...

A simple random sample of size n is drawn. The sample​ mean, x overbarx​, is found to be 18.5​, and the sample standard​ deviation, s, is found to be 4.3.

(a) Construct a 95​% confidence interval about muμ if the sample​ size, n, is 34.

Lower​ bound: ____

Upper​ bound: ____

​(b) Construct a 95​% confidence interval about muμ if the sample​ size, n, is 51.

Lower​ bound: ____

Upper​ bound: ____

​(c) Construct a 99​% confidence interval about muμ if the sample​ size, n, is 34.

Lower​ bound: ____

Upper​ bound: ____

Solutions

Expert Solution

Solution :

Given that,

a)

Point estimate = sample mean = = 18.5

sample standard deviation = s = 4.3

sample size = n = 34

Degrees of freedom = df = n - 1 = 33

At 95% confidence level the t is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

t /2,df = t0.025,33 = 2.035

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

= 2.035 * ( 4.3 / 34)

= 1.501

The 95% confidence interval estimate of the population mean is,

- E < < + E

18.5 - 1.501 < < 18.5 + 1.501

16.999 < < 20.001

Lower bound = 16.999

Upper bound = 20.001

b)

sample size = n = 51

Degrees of freedom = df = n - 1 = 50

At 95% confidence level the t is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

t /2,df = t0.025,50 = 2.009

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

= 2.009 * ( 4.3 / 51)

= 1.210

The 95% confidence interval estimate of the population mean is,

- E < < + E

18.5 - 1.210 < < 18.5 + 1.210

17.29 < < 19.71

Lower bound = 17.29

Upper bound = 19.71

c)

sample size = n = 34

Degrees of freedom = df = n - 1 = 33

At 99% confidence level the t is ,

= 1 - 99% = 1 - 0.99 = 0.01

/ 2 = 0.01 / 2 = 0.005

t /2,df = t0.005,33 = 2.733

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

= 2.733 * ( 4.3 / 34)

= 2.015

The 99% confidence interval estimate of the population mean is,

- E < < + E

18.5 - 2.015 < < 18.5 + 2.015

16.485 < < 20.515

Lower bound = 16.485

Upper bound = 20.515


Related Solutions

A simple random sample of size n is drawn. The sample​ mean, x overbarx​, is found...
A simple random sample of size n is drawn. The sample​ mean, x overbarx​, is found to be 18.618.6​, and the sample standard​ deviation, s, is found to be 4.84.8. LOADING... Click the icon to view the table of areas under the​ t-distribution. ​(a) Construct a 9595​% confidence interval about muμ if the sample​ size, n, is 3434. Lower​ bound: nothing​; Upper​ bound: nothing ​(Use ascending order. Round to two decimal places as​ needed.) ​(b) Construct a 9595​% confidence interval...
A simple random sample of size n is drawn. The sample​ mean, x​, is found to...
A simple random sample of size n is drawn. The sample​ mean, x​, is found to be 18.5, and the sample standard​ deviation, s, is found to be 4.6. ​(a) Construct a 95​% confidence interval about μ if the sample​ size, n, is 34. Lower​ bound: ___ Upper​ bound: ___ ​(Use ascending order. Round to two decimal places as​ needed.) ​(b) Construct a 95% confidence interval about μ if the sample​ size, n, is 61. Lower​ bound: ___ Upper​ bound:...
A simple random sample of size n is drawn. The sample mean, x, is found to...
A simple random sample of size n is drawn. The sample mean, x, is found to be 35.1, and the sample standard deviation, s, is found to be 8.7 a) Construct a 90% confidence interval for μ if the sample size, n, is 100. b) Construct a 90% confidence interval for μ if the sample size, n, is 40. How does decreasing the sample size affect the margin of error, E? c) Construct a 96% confidence interval for μ if...
A simple random sample of size n is drawn. The sample​ mean, x ​, is found...
A simple random sample of size n is drawn. The sample​ mean, x ​, is found to be 18.5 ​, and the sample standard​ deviation, s, is found to be 4.3. (a) Construct a 95​% confidence interval about mean if the sample​ size, n, is 35. (b) Construct a 95% confidence interval about mean if the sample​ size, n, is 71. How does increasing the sample size affect the margin of​ error, E? ​(c) Construct a 99​% confidence interval about...
A simple random sample of size n is drawn. The sample​ mean, x is found to...
A simple random sample of size n is drawn. The sample​ mean, x is found to be 19.3 and the sample standard​ deviation, s, is found to be 4.7 a. Construct a 95% confidence interval about mu if the sample​ size, n, is 35
A simple random sample of size n is drawn. The sample​ mean, x overbar​, is found...
A simple random sample of size n is drawn. The sample​ mean, x overbar​, is found to be 18.4​, and the sample standard​ deviation, s, is found to be 4.8. Construct a 95​% confidence interval about μ if the sample​ size, n, is 34.
A simple random sample of size n is drawn. The sample​ mean, x overbar​, is found...
A simple random sample of size n is drawn. The sample​ mean, x overbar​, is found to be 18.9​, and the sample standard​ deviation, s, is found to be 4.4. a) Construct a 95​% confidence interval about mu if the sample​ size, n, is 34. Lower​ bound: nothing​; Upper​ bound: nothing ​(Use ascending order. Round to two decimal places as​ needed.) ​(b) Construct a 95​% confidence interval about mu if the sample​ size, n, is 61. Lower​ bound: nothing​; Upper​...
A simple random sample of size n is drawn. The sample​ mean is found to be...
A simple random sample of size n is drawn. The sample​ mean is found to be 17.6​, and the sample standard​ deviation, s, is found to be 4.7. A). Construct a 95​% confidence interval about if the sample​ size, n, is 51.
A simple random sample of size n is drawn. The sample​ mean is found to be...
A simple random sample of size n is drawn. The sample​ mean is found to be 17.6​, and the sample standard​ deviation, s, is found to be 4.7. Construct a 99​% confidence interval about if the sample​ size, n, is 34.
A simple random sample of size n is drawn. The sample​ mean, x overbar x​, is...
A simple random sample of size n is drawn. The sample​ mean, x overbar x​, is found to be 17.8​, and the sample standard​ deviation, s, is found to be 4.9. ​(a) Construct a 95​% confidence interval about mu μ if the sample​ size, n, is 34. Lower​ bound= Upper​ bound= ​(Use ascending order. Round to two decimal places as​ needed.) ​(b) Construct a 95​% confidence interval about mu μ if the sample​ size, n, is 61. Lower​ bound=​ Upper​...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT