A simple random sample of 11 observations from a population containing 400 female soccer players was taken, and the following values were obtained. 48 53 72 56 63 64 56 76 50 46 73Calculate a 95% confidence interval for the population mean.
Question
A simple random sample of 11 observations from a population containing 400 female soccer players was taken, and the following values were obtained. 48 53 72 56 63 64 56 76 50 46 73Calculate a 95% confidence interval for the population mean.
Solution
Sure, here are the steps to calculate a 95% confidence interval for the population mean:
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First, calculate the sample mean (x̄). This is done by adding all the values together and then dividing by the number of observations. In this case, the sum of the values is 657 and there are 11 observations, so the sample mean is 657 / 11 = 59.73.
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Next, calculate the sample standard deviation (s). This is done by subtracting each value from the mean, squaring the result, adding all these squared values together, dividing by the number of observations minus 1, and then taking the square root of the result. The calculation is a bit complex, but the standard deviation for this data set is approximately 10.74.
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Now, look up the critical value (z*) for a 95% confidence interval in a standard normal distribution table. This value is approximately 1.96.
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Finally, calculate the margin of error (E) using the formula E = z* * (s / √n), where n is the number of observations. In this case, E = 1.96 * (10.74 / √11) = 6.32.
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The 95% confidence interval is then x̄ ± E, or 59.73 ± 6.32. Therefore, we can be 95% confident that the population mean is between 53.41 and 66.05.
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