To calculate the mean and standard deviation for a sample, sum the values and divide by the number of samples; for standard deviation, use the squared differences from the mean. Comparisons and hypotheses can be made based on the calculated sample statistics and known population parameters.
To find the mean and standard deviation for a given sample, you can follow these basic steps:
Calculate the mean (average) by summing all the sample values and then dividing by the number of samples.
For standard deviation, first find the difference between each sample value and the mean, square each difference, sum these squares, divide by the number of samples (for a population) or by the number of samples minus one (for a sample), and then take the square root of the result.
Given that the mean of the original distribution is provided along with the standard deviation of the original distribution and a sample size (n), these statistics can be used for further calculations such as probabilities and hypothesis testing.
For the candy company's problem, the sample mean weight, standard deviation of the sample weights, and sum of the sample weights can be found using the data from the randomly selected candies. To determine the accuracy of the candy company's labeling, one could compare the sample statistics to the company's stated mean and standard deviation.
When dealing with quality control and comparing means between two groups (for example, chocolate candies and peanut butter candies), a hypothesis test can be conducted. If standard deviations are known and population distributions are normal, a test of means or proportions can be performed to compare the average values.