Question

A. Find the value of Beta, when the Null Hypothesis assumes a population mean of Mu = 950, with a population standard deviation of 180, the sample size is 4 and the true mean is 1087.87 with confidence interval of 95%

B. Find the value of Beta, when the Null Hypothesis assumes a population mean of Mu = 900, with a population standard deviation of 277, the sample size is 7 and the true mean is 1202.94 with confidence interval of 95%

C. Find the value of Beta, when the Null Hypothesis assumes a population mean of Mu = 600, with a population standard deviation of 142, the sample size is 12 and the true mean is 744.04 with confidence interval of 95% PLEASE SHOW WORK

Answer 1

A.

Standard error of mean = = 180 / = 90

Z value for 95% confidence interval is 1.96

For confidence interval of 95%, the critical values to reject the null hypothesis are,

950 - 1.96 * 90 and 950 + 1.96 * 90

773.6 and 1126.4

We reject H0 if sample mean is less than 773.6 or greater than 1126.4

Beta = P(Accept H0 | = 1087.87)

= P(773.6 < < 1126.4 | = 1087.87)

= P[ < 1126.4 | = 1087.87] - P[ < 773.6 | = 1087.87]

= P[Z < (1126.4 - 1087.87) /90] - P[Z < (773.6 - 1087.87) /90]

= P[Z < 0.4281] - P[Z < -3.4919]

= 0.6657 - 0.0002

= 0.6655

B.

Standard error of mean = = 277 / = 104.6962

Z value for 95% confidence interval is 1.96

For confidence interval of 95%, the critical values to reject the null hypothesis are,

900 - 1.96 * 104.6962 and 900 + 1.96 * 104.6962

694.7954 and 1105.205

We reject H0 if sample mean is less than 694.7954 or greater than 1105.205

Beta = P(Accept H0 | = 1202.94)

= P(694.7954 < < 1105.205 | = 1202.94)

= P[ < 1105.205 | = 1202.94] - P[ < 694.7954 | = 1202.94]

= P[Z < (1105.205 - 1202.94) /104.6962] - P[Z < (694.7954 - 1202.94) /104.6962]

= P[Z < -0.9335] - P[Z < -4.8535]

= 0.1752 - 0

= 0.1752

C.

Standard error of mean = = 142 / = 40.99187

Z value for 95% confidence interval is 1.96

For confidence interval of 95%, the critical values to reject the null hypothesis are,

600 - 1.96 * 40.99187 and 600 + 1.96 * 40.99187

519.6559 and 680.3441

We reject H0 if sample mean is less than 519.6559 or greater than 680.3441

Beta = P(Accept H0 | = 744.04)

= P(519.6559 < < 680.3441 | = 744.04)

= P[ < 680.3441 | = 744.04] - P[ < 519.6559 | = 744.04]

= P[Z < (680.3441 - 744.04) /40.99187] - P[Z < (519.6559 - 744.04) /40.99187]

= P[Z < -1.5539] - P[Z < -5.4739]

= 0.0601 - 0

= 0.0601