An AC Circuit has its frequency resonance measured in 30 times. The mean of the measured data is 2160 Hertz and the standard deviation is 88 Hertz. The data is normally distributed. Find the 70%, 80%, and the 90% confidence interval for these population mean. Also, comment on the number of measurements expected to lie within the intervals.
Solutions
Given that;
The formula for determining the confidence interval (C.I) is given as:
1
For 70% confidence interval, the value of is 1.04
Therefore, the 70% confidence interval is (2089.29, 2122.71)
For 80% confidence interval, the value of is 1.28
Therefore, the 80% confidence interval is (2085.43, 2126.56)
For 90% confidence interval, the value of is 1.645
Therefore, the 90% confidence interval is (2079.57, 2132.42).
Comment
Confidence limits are larger if a higher level of confidence is selected, that is 70% compared with the 80 and 90%). If the experiment is repeated, the mean possibly will fall within the C.I limits.
In a vacuum chamber, the pressure is measured in ten days and the data recoded as shown in the table below. Use T-distribution to calculate 95% confidence interval for the population mean of these data.
Solution
Given that:
Sample size,
Standard deviation,
The degree of freedom,
And the T-value at 95% confidence interval with d.f = 9 is
Then the confidence interval for t-distribution is represented as:
2
Thus the confidence interval is (131.29, 143.51)
Measurements can be taken by the use of a meter rule up to a precision of one millimeter. But a prolonged use of the rule exposes it to a number of atmospheric conditions, that is, varying temperatures and humidity that could cause the wood to shrink or/and expend during the measurements. These effects lead to systematic errors into the measurement of the length such that the estimated uncertainty is more than a millimeter. A wooden meter rule was used to measure the image position formed by lens in an optics lab. In the experiment, eight measurements were taken and recorded in the table below. Use the table to find:
Mean image of the distance
Estimated standard deviation
Mean standard error
The 95 percent confidence interval of the distance of the image without the systematic errors. Take into account that only the eight measurements were taken.
The 95 percent confidence interval of the distance of the image without the random errors. Take into account that only the eight measurements were taken.
The 95 percent confidence interval of the distance of the image with random the systematic errors. Take into account that only the eight measurements were taken.
Solution
The mean of the data is determined by:
S = 1.98
95 percent confidence interval of the distance of the image without the systematic errors
95 percent confidence interval of the distance of the image without the random errors.
95 percent confidence interval of the distance of the image with random the systematic errors
