Answer :
We're 90% confident that the true population mean lies somewhere between 67.73 and 69.38, indicating high precision and reliability of our estimate. The calculations are made to two decimal places.
Firstly, we start by calculating the sample mean. The sample mean is simply the average of all the given data values. In this case the sample mean comes out to be approximately 68.55 which represents the central tendency of the sample.
Next, we need to calculate the standard error. The standard error gives us an idea of the variability of the sample mean, and tells us how accurate our estimate of the population mean is likely to be. It's calculated by dividing the population standard deviation (which is given as 2.89) by the square root of the sample size. After performing the calculation, we get the standard error approximately equal to 0.50.
Once we know the standard error, we can then find our z-score for a 90% confidence interval. This z-score gives us an idea of how many standard deviations away from the mean we expect our interval to be. For a 90% confidence interval, this z-score happens to be approximately 1.64.
With the z-score in hand, we can now compute the boundaries of our confidence interval. A confidence interval gives us a range of values in which we're confident that the true population parameter (in this case, the mean) is likely to lie.
The lower bound of the confidence interval is calculated by subtracting the product of the z-score and standard error from the sample mean, and the upper bound is calculated by adding the product of the z-score and standard error to the sample mean. Here, we get our confidence interval of approximately (67.73, 69.38).
This tells us that we're 90% confident that the true population mean lies somewhere between 67.73 and 69.38, indicating high precision and reliability of our estimate. The calculations are made to two decimal places.
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