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The sample mean, symbolized as \(\bar{x}\), is a critical value that represents the average of a subset of the population being studied. In our exercise, the sample mean was calculated to be 149 based on the responses of 50 randomly chosen Facebook users.

Understanding the sample mean is essential because it serves as an estimator of the population mean. To calculate it, you simply sum up all the values in the sample and then divide the sum by the number of observations in the sample. It should be noted that while the sample mean can give a good estimation of the population mean, it is not necessarily an exact match due to the variability inherent in sampling.

The accuracy of the sample mean as an estimator of the population mean increases with the sample size. Larger samples tend to give a mean value that is closer to the population mean, which illustrates the concept of the Law of Large Numbers. The reliability of conclusions drawn from the sample mean can be assessed through statistical tests such as the calculation of a z-statistic.

The population mean, represented by the Greek letter \(\mu\), refers to the average of all individuals or elements in the entire group we aim to analyze or understand. It is a parameter that signifies the central point of the distribution of the entire population.

The term might sound similar to the sample mean, but the key difference lies in the scope: the population mean considers every single member of the group without exception. In our specific exercise scenario, the reported population mean of Facebook friends is 155.

Obtaining the exact population mean can be challenging or even impractical when dealing with large populations, which is why researchers often rely on samples to estimate this value. It is the true average we are trying to estimate or get close to by collecting and analyzing sample data.

The sample standard deviation (commonly denoted as \(s\)) is a measure of the amount of variability or spread in a sample data set. It quantifies how much the individual data points differ from the sample mean.

In our exercise, the sample standard deviation is 29.7, which suggests that on average, the number of friends that each individual in the sample has deviates from the sample mean (149) by about 29.7. Calculating the standard deviation involves determining the average squared deviation of each data point from the sample mean, taking the square root of that average.

A higher standard deviation indicates that the data points are more spread out from the mean, whereas a lower standard deviation implies that they are clustered closer to the mean. Importantly, while the standard deviation provides insights into the variability within a sample, it does not impact the sample mean itself.

The concept of statistical significance relates to the likelihood that the result obtained from a data sample can be generalized to the population at large. It indicates whether any observed differences or relationships in the data are meaningful or if they could simply be due to random chance.

In terms of the z-statistic, absolute values of z greater than 2 typically indicate statistical significance, suggesting that the sample mean is different from the population mean to a degree that is unlikely to occur by random chance. In the given exercise, the computed z-value of approximately -1.43 does not exceed the standard threshold of 2, indicating that the difference between our sample mean and population mean is not statistically significant.

This means that if there is a true difference in the average number of Facebook friends, the results from this specific sample do not sufficiently show it. It's important to consider that 'not significant' does not imply 'no difference'; rather, it points to insufficient evidence to conclude that a difference exists based on the sample data.