Answer :
The estimated error of the mean (standard error) can be calculated by dividing the standard deviation of the sample by the square root of the sample size.
However, the standard deviation of the population is unknown, so we use the sample standard deviation as an estimate.
Therefore, the estimated error of the mean is the sample standard deviation divided by the square root of the sample size.
To calculate the estimated error of the mean (standard error), we need to find the sample standard deviation and the square root of the sample size. Let's first calculate the sample standard deviation:
1. Calculate the mean (average) of the data:
Mean = (76.3 + 84 + 81.5 + 95.8 + 98.5 + 92.5 + 94 + 87.4 + 94.6 + 86.7) / 10 = 89.8
2. Calculate the differences between each data point and the mean, and square them:
(76.3 - 89.8)^2 = 161.29
(84 - 89.8)^2 = 34.81
(81.5 - 89.8)^2 = 68.89
(95.8 - 89.8)^2 = 36
(98.5 - 89.8)^2 = 75.69
(92.5 - 89.8)^2 = 7.29
(94 - 89.8)^2 = 17.64
(87.4 - 89.8)^2 = 5.76
(94.6 - 89.8)^2 = 23.04
(86.7 - 89.8)^2 = 9.61
3. Sum up the squared differences:
161.29 + 34.81 + 68.89 + 36 + 75.69 + 7.29 + 17.64 + 5.76 + 23.04 + 9.61 = 459.92
4. Divide the sum by (n - 1), where n is the sample size (10 in this case):
Variance = 459.92 / (10 - 1) = 51.1
5. Take the square root of the variance to find the sample standard deviation:
Sample Standard Deviation = √(51.1) ≈ 7.14
Now that we have the sample standard deviation, we can calculate the estimated error of the mean (standard error) by dividing it by the square root of the sample size:
Estimated Error of the Mean (Standard Error) = Sample Standard Deviation / √(Sample Size) = 7.14 / √(10) ≈ 2.26
Therefore, the estimated error of the mean (standard error) is approximately 2.26.
The estimated error of the mean, also known as the standard error, quantifies the variability or uncertainty associated with the sample mean as an estimate of the population means.
It indicates how much the sample mean is likely to deviate from the true population mean.
In this case, we calculated the estimated error of the mean to be approximately 2.26.
This means that if we were to repeatedly sample from the population and calculate the mean, we would expect the sample means to vary around the true population mean by about 2.26 units on average.
To estimate a confidence interval for the population mean, we can use the formula:
Confidence Interval = Sample Mean ± (Critical Value) × (Standard Error)
Assuming a 95% confidence interval
Learn more about Confidence Interval
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