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Hypothesis testing is a statistical method used to make inferences or decisions about a population parameter based on sample data. In this exercise, we are testing the variance of solar battery lifetimes, which affects how long the satellite can operate effectively.
The process begins by clearly stating the hypotheses:

• Null Hypothesis (\(H_0\)): \(\sigma^2 = 23\) — This suggests that the actual variance of the battery lifetimes is 23 months squared, as desired by the engineers.
• Alternative Hypothesis (\(H_a\)): \(\sigma^2 eq 23\) — This suggests there is a difference from the expected variation, which could be either more or less than 23 months squared.

Next, we choose the appropriate test, which is a chi-square test in this case, because it is used for variance testing. By calculating the chi-square statistic and comparing it against critical values from the chi-square distribution, we decide whether to reject the null hypothesis. A key step involves determining degrees of freedom, which is \(n - 1\), with \(n\) being the sample size.
Ultimately, the hypothesis test concluded that there was not enough evidence to reject the null hypothesis — indicating the variance might indeed be close to the desired 23 months squared.

A confidence interval gives an estimated range of values likely to include a population parameter, in this case, the population variance. Calculating a confidence interval for the variance helps provide a range within which the true variance is expected to fall, with a specific level of confidence.

For this task, the steps involve determining a 90% confidence interval for both the population variance and the standard deviation.
The formula for the confidence interval of the variance is:\[\left(\frac{(n-1)s^2}{\chi^2_{upper}}, \frac{(n-1)s^2}{\chi^2_{lower}}\right)\] Here, \(\chi^2_{lower}\) and \(\chi^2_{upper}\) are critical values obtained from the chi-square distribution.
This results in an interval providing a range for the variance: \((9.19, 25.88)\).
This method can then be extended to find the confidence interval for the standard deviation by taking the square root of these interval bounds. This way, we are 90% confident that the actual variance is between 9.19 and 25.88, and the standard deviation is between 3.03 and 5.08 months.

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In the context of this exercise, understanding the standard deviation helps assess how spread out the battery lifetimes are, influencing satellite performance.

Standard deviation is derived directly from the variance, which in mathematical terms is the square root of the variance:\[\sigma = \sqrt{\sigma^2}\]
In this given scenario, once the confidence interval for the variance was determined as \((9.19, 25.88)\), the confidence interval for the standard deviation was calculated by taking the square root of these numbers. This resulted in a 90% confidence interval of \((3.03, 5.08)\) for the standard deviation.
Such a range provides insight into how much individual battery lifetimes deviate from the average, which is crucial for predicting the satellite's operational consistency and planning for battery replacements or maintenance strategies.

Sample variance is a statistic that provides a measure of how much the data values in a sample deviate from the mean of the sample. It is essential for estimating the variance of a population when only a subset of data is available.

In calculating sample variance, each data point's deviation from the mean is squared and averaged over the sample size minus one (\(n-1\)), providing an unbiased estimator of the population variance. This bias correction is necessary due to the finite sample size.
The formula for sample variance \(s^2\) is:\[s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}\] where \(x_i\) are individual data points and \(\bar{x}\) is the sample mean.
In the provided exercise, the sample variance was calculated as \(14.3\) months squared. This serves as the baseline statistic used to test hypotheses about the overall battery variance and to calculate confidence intervals.
Employing sample variance is crucial in ensuring that decisions and predictions based on sample data are as accurate as possible.