Small related probability problems SHow full work please Ref

Small related probability problems. SHow full work please

Refer to chapter 8 of the text. Heights (cm) of a random sample of students at a small Midwestern college: 183 154 170 189 187 176 193 150 181 187 162 175 157 159 185 164 189 173 166 179 157 177 169 159 188 191 163 183 177 180 Use these data for problems on this task. 1. Find point estimates of p and o. (1) 2. Assuming sigma = 12 cm. construct a 95% confidence interval for mu . (1) 3. Find the margin of error and width of the confidence interval constructed in problem 2. (1) 4 What sample size is required to reduce the margin of error in problem 3 by 25%? by 50%? (1) 5. Assume sigma is unknown. Construct a 95% confidence interval for p. (1)

Solution

1)
The point of estimate for mean(u) = Average of the 30 observation calculated = 173.767

2)

CI = x ± Z a/2 * (sd/ Sqrt(n))
Where,
x = Mean
sd = Standard Deviation
a = 1 - (Confidence Level/100)
Za/2 = Z-table value
CI = Confidence Interval
Mean(x)=173.767
Standard deviation( sd )=12
Sample Size(n)=30
Confidence Interval = [ 173.767 ± Z a/2 ( 12/ Sqrt ( 30) ) ]
= [ 173.767 - 1.96 * (2.191) , 173.767 + 1.96 * (2.191) ]
= [ 169.473,178.061 ]

3)
Margin of Error = Z a/2 * (sd/ Sqrt(n))
Where,
x = Mean
sd = Standard Deviation
a = 1 - (Confidence Level/100)
Za/2 = Z-table value
Mean(x)=173.767
Standard deviation( sd )=12
Sample Size(n)=30
Margin of Error = Z a/2 * 12/ Sqrt ( 30)
= 1.96 * (2.191)
= 4.294

4) WHEN Population standard deviation is unknown
CI = x ± t a/2 * (sd/ Sqrt(n))
Where,
x = Mean
sd = Standard Deviation
a = 1 - (Confidence Level/100)
ta/2 = t-table value
CI = Confidence Interval
Mean(x)=173.767
Standard deviation( sd )=12.8967
Sample Size(n)=30
Confidence Interval = [ 173.767 ± t a/2 ( 12.8967/ Sqrt ( 30) ) ]
= [ 173.767 - 2.0452 * (2.355) , 173.767 + 2.0452 * (2.355) ]
= [ 168.951,178.583 ]