The mean of a population is a measure of the central tendency, representing the average value of a dataset. In this case, the population has a mean of 80, which indicates that if you were to sum all the individual observations in the population and then divide by the number of observations, you would arrive at 80.
The standard deviation, which is given as 7, measures the amount of variation or dispersion in the population. A standard deviation of 7 means that most of the data points in the population lie within 7 units above or below the mean (80). This gives us an idea of how spread out the values are around the mean.
When we take a sample of 49 observations from this population, we can use the same mean and standard deviation to understand how our sample might behave. Given that our sample size is large (n=49), we can apply the Central Limit Theorem, which states that the sampling distribution of the sample mean will be approximately normally distributed, regardless of the population’s distribution, provided the sample size is sufficiently large.
In our case, with a mean of 80 and a standard deviation of 7, we can also calculate the standard error of the mean (SEM), which is the standard deviation of the sample mean. The SEM can be calculated using the formula:
SEM = σ / √n
Where σ is the population standard deviation and n is the sample size. Substituting the values:
SEM = 7 / √49 = 7 / 7 = 1
This means that the sample mean will fluctuate around the population mean (80) by about 1 unit. This small standard error indicates that we can be quite confident about the sample mean being close to the population mean, given the sizable sample size. In summary, understanding the mean and standard deviation provides a solid foundation for interpreting the characteristics of our sample in relation to the overall population.