A sample has the following characteristics :
- Size : n=64
- Standard deviation : \sigma=2
- Mean : \bar{x}=13
Find a confidence interval for the population mean with a confidence of 95%.
The confidence interval when the standard deviation is known can be computed using the following formula:
\left[ \overline{x}-z^*\dfrac{\sigma}{\sqrt{n}},\overline{x}+z^*\dfrac{\sigma}{\sqrt{n}} \right]
According to the z -table, the value of z^* corresponding to confidence of 95 percent is 1.96. So we have:
z^*\dfrac{\sigma}{\sqrt{n}} = 1.96\dfrac{2}{\sqrt{64}}=0.49
The confidence interval is:
\left[ 13- 0.49 , 13 + 0.49\right]
Or:
\left[ 12.51, 13.49 \right]
The population mean has a 95% chance to be on \left[12.51{,}13.49\right]
A sample has the following characteristics :
- Size : n=100
- Standard deviation : \sigma=18
- Mean : \bar{x}=86
Find a confidence interval for the population mean with a confidence of 95%.
The confidence interval when the standard deviation is known can be computed using the following formula:
\left[ \overline{x}-z^*\dfrac{\sigma}{\sqrt{n}},\overline{x}+z^*\dfrac{\sigma}{\sqrt{n}} \right]
According to the z -table, the value of z^* corresponding to confidence of 95 percent is 1.96. So we have:
z^*\dfrac{\sigma}{\sqrt{n}} = 1.96\dfrac{18}{\sqrt{100}}=3.53
The confidence interval is:
\left[ 86-3.53, 86+3.53\right]
Or:
\left[ 82.47{,}89.53 \right]
The population mean has a 95% chance to be on \left[ 82.47{,}89.53 \right]
A sample has the following characteristics :
- Size : n=625
- Standard deviation : \sigma=50
- Mean : \bar{x}=145
Find a confidence interval for the population mean with a confidence of 99%.
The confidence interval when the standard deviation is known can be computed using the following formula:
\left[ \overline{x}-z^*\dfrac{\sigma}{\sqrt{n}},\overline{x}+z^*\dfrac{\sigma}{\sqrt{n}} \right]
According to the z -table, the value of z^* corresponding to confidence of 99 percent is 2.58. So we have:
z^*\dfrac{\sigma}{\sqrt{n}} = 2.58\dfrac{50}{\sqrt{625}}= 5.16
The confidence interval is:
\left[ 145-5.16, 145+5.16\right]
Or:
\left[ 139.84{,}150.16 \right]
The population mean has a 99% chance to be on \left[ 139.84{,}150.16 \right]
A sample has the following characteristics :
- Size : n=289
- Standard deviation : \sigma=68
- Mean : \bar{x}=401
Find a confidence interval for the population mean with a confidence of 95%.
The confidence interval when the standard deviation is known can be computed using the following formula:
\left[ \overline{x}-z^*\dfrac{\sigma}{\sqrt{n}},\overline{x}+z^*\dfrac{\sigma}{\sqrt{n}} \right]
According to the z -table, the value of z^* corresponding to confidence of 95 percent is 1.96. So we have:
z^*\dfrac{\sigma}{\sqrt{n}} = 1.96\dfrac{68}{\sqrt{289}}=7.84
The confidence interval is:
\left[ 401-7.84, 401+7.84\right]
Or:
\left[ 393.16{,}408.84 \right]
The population mean has a 95% chance to be on \left[ 393.16{,}408.84 \right]
A sample has the following characteristics :
- Size : n=324
- Standard deviation : \sigma=18
- Mean : \bar{x}=99
Find a confidence interval for the population mean with a confidence of 95%.
The confidence interval when the standard deviation is known can be computed using the following formula:
\left[ \overline{x}-z^*\dfrac{\sigma}{\sqrt{n}},\overline{x}+z^*\dfrac{\sigma}{\sqrt{n}} \right]
According to the z -table, the value of z^* corresponding to confidence of 95 percent is 1.96. So we have:
z^*\dfrac{\sigma}{\sqrt{n}} = 1.96\dfrac{18}{\sqrt{324}}=1.96
The confidence interval is:
\left[ 99-1.96{,}99+1.96\right]
Or:
\left[97.04, 100.96 \right]
The population mean has a 95% chance to be on \left[97.04, 100.96 \right]
A sample has the following characteristics :
- Size : n=81
- Standard deviation : \sigma= 27
- Mean : \bar{x}=55
Find a confidence interval for the population mean with a confidence of 90%.
The confidence interval when the standard deviation is known can be computed using the following formula:
\left[ \overline{x}-z^*\dfrac{\sigma}{\sqrt{n}},\overline{x}+z^*\dfrac{\sigma}{\sqrt{n}} \right]
According to the z -table, the value of z^* corresponding to confidence of 90 percent is 1.65. So we have:
z^*\dfrac{\sigma}{\sqrt{n}} = 1.65\dfrac{27}{\sqrt{81}}=4.95
The confidence interval is:
\left[ 55-4.95, 55+4.95\right]
Or:
\left[ 50.05{,}59.95 \right]
The population mean has a 90% chance to be on \left[ 50.05{,}59.95 \right]
A sample has the following characteristics :
- Size : n=256
- Standard deviation : \sigma=32
- Mean : \bar{x}=75
Find a confidence interval for the population mean with a confidence of 99%.
The confidence interval when the standard deviation is known can be computed using the following formula:
\left[ \overline{x}-z^*\dfrac{\sigma}{\sqrt{n}},\overline{x}+z^*\dfrac{\sigma}{\sqrt{n}} \right]
According to the z -table, the value of z^* corresponding to confidence of 99 percent is 2.58. So we have:
z^*\dfrac{\sigma}{\sqrt{n}} = 2.58\dfrac{32}{\sqrt{256}}=5.16
The confidence interval is:
\left[ 75-5.16, 75+5.16\right]
Or:
\left[ 69.84{,}80.16 \right]
The population mean has a 99% chance to be on \left[ 69.84{,}80.16 \right]