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The sample mean, often represented as \( \bar{x} \), is a key statistic that represents the average value of a sample. In Michelson's experiment, where he measured the speed of light 100 times, the sample mean is given as \( 299,852.4 \; \text{km/s} \). This mean is calculated by adding up all the measured values and dividing by the sample size, which is 100 in this case.

The sample mean provides an estimate of the true population mean, which is what scientists aim to measure. However, it is important to remember that this is an estimate based on the sample and can differ from the true population mean. The reliability of the sample mean as an estimate improves with larger sample sizes.

Standard deviation, represented by \( s \), measures the amount of variation or dispersion in a set of values. In Michelson's measurements, the standard deviation is \( 79.01 \; \text{km/s} \). This tells us that, on average, the individual measurements are about 79.01 km/s away from the sample mean.

It's an important statistic because it provides insight into the consistency of the data. A smaller standard deviation means that the data points are closer to the mean, indicating more reliable measurements. Conversely, a larger standard deviation suggests more variability, which can influence our confidence in the sample mean as a representation of the true mean.

The standard error (SE) of the mean measures how much the sample mean is expected to vary from the actual population mean. It is calculated using the formula \( SE = \frac{s}{\sqrt{n}} \), where \( s \) is the standard deviation and \( n \) is the sample size.

In Michelson's case, the standard error is \( 7.901 \; \text{km/s} \). The standard error is crucial because it helps in constructing the confidence interval, which indicates how accurate the sample mean is likely to be as a representation of the population mean. A smaller SE suggests that the sample mean is a more precise estimate of the true mean.

The critical value is a key component when constructing confidence intervals. For a 95% confidence interval with a large sample size, such as Michelson's 100 measurements, the critical value from the standard normal distribution is approximately 1.96.

This value indicates how many standard errors away from the sample mean we need to go to capture the true population mean with the desired confidence level. Multiplying the critical value by the standard error gives us the margin of error. The larger the critical value, the wider the confidence interval, which reflects more uncertainty in the estimate. Thus, understanding the critical value is essential for interpreting the range where the true mean is likely to lie.