The arsenic concentration in soil is measured 7 times. The results showed that the mean was 4 ppm with a standard deviation of 0.09 ppm. The standard allowable limit for arsenic in soil is 2 ppm. Do a hypothesis testing for the data at 95% confidence level.
Data:
Hypothesis:
Test statistic:
Decision Rule:
Decision:
Answers
Standard deviation in statistics, typically denoted by σ, is a measure of variation or dispersion (refers to a distribution's extent of stretching or squeezing) between values in a set of data. The lower the standard deviation, the closer the data points tend to be to the mean (or expected value), μ. Conversely, a higher standard deviation indicates a wider range of values. Similarly to other mathematical and statistical concepts, there are many different situations in which standard deviation can be used, and thus many different equations. In addition to expressing population variability, the standard deviation is also often used to measure statistical results such as the margin of error. When used in this manner, standard deviation is often called the standard error of the mean, or standard error of the estimate with regard to a mean. The calculator above computes population standard deviation and sample standard deviation, as well as confidence interval approximations.
Population Standard Deviation
The population standard deviation, the standard definition of σ, is used when an entire population can be measured, and is the square root of the variance of a given data set. In cases where every member of a population can be sampled, the following equation can be used to find the standard deviation of the entire population:
Where
xi is an individual value
μ is the mean/expected value
N is the total number of values
For those unfamiliar with summation notation, the equation above may seem daunting, but when addressed through its individual components, this summation is not particularly complicated. The i=1 in the summation indicates the starting index, i.e. for the data set 1, 3, 4, 7, 8, i=1 would be 1, i=2 would be 3, and so on. Hence the summation notation simply means to perform the operation of (xi - μ2) on each value through N, which in this case is 5 since there are 5 values in this data set.
EX: μ = (1+3+4+7+8) / 5 = 4.6
σ = √[(1 - 4.6)2 + (3 - 4.6)2 + ... + (8 - 4.6)2)]/5
σ = √(12.96 + 2.56 + 0.36 + 5.76 + 11.56)/5 = 2.577
Answer: