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Answer :
The answer for the following are: i) sample mean = 5.1286 ii) sample variance = 0.1683 iii) The 90% confidence interval for the population mean based on the given data is approximately [4.8646, 5.3926].
Given 7 independent observations of a normal variable with the sum of x-values (Σx) equal to 35.9 and the sum of squared x-values (∑x^2) equal to 186.19, we can calculate the sample mean, sample variance, and a 90% confidence interval for the population mean.
i. To calculate the sample mean, we divide the sum of x-values (Σx) by the number of observations (n): sample mean = Σx / n. In this case, the sample mean is 35.9 / 7 = 5.1286.
ii. To calculate the sample variance, we first need to find the sum of squares (Σx^2) using the formula Σx^2 = ∑(x^2). Then, we can use the formula for sample variance: sample variance = (Σx^2 - (Σx)^2 / n) / (n - 1). Plugging in the given values, we have (186.19 - (35.9)^2 / 7) / (7 - 1) = 0.1683.
iii. To calculate a 90% confidence interval for the population mean, we can follow these steps:
Calculate the sample mean (X) by dividing the sum of x-values (Σx) by the number of observations (n). In this case, X = 35.9 / 7 = 5.1286.
Calculate the sample variance (s^2) using the formula (Σx^2 - (Σx)^2 / n) / (n - 1). Substituting the given values, we have s^2 = (186.19 - (35.9)^2 / 7) / (7 - 1) = 0.1683.
Calculate the standard error (SE) by taking the square root of the sample variance divided by the square root of the sample size: SE = sqrt(s^2 / n) = sqrt(0.1683 / 7) = 0.1359.
Determine the critical value corresponding to the desired confidence level. Since the sample size is small (n = 7), we use the t-distribution instead of the z-distribution. With 6 degrees of freedom (n - 1), a 90% confidence level corresponds to a critical t-value of approximately 1.943.
Calculate the margin of error (ME) by multiplying the critical value by the standard error: ME = t * SE = 1.943 * 0.1359 = 0.2640.
Finally, construct the confidence interval by subtracting and adding the margin of error to the sample mean: Confidence Interval = X ± ME = 5.1286 ± 0.2640.
Therefore, the 90% confidence interval for the population mean based on the given data is approximately [4.8646, 5.3926]. This means that we can be 90% confident that the true population mean falls within this range.
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