A random sample size 15 from a normal population yields a mean value of 2.23 and a variance of 7.33.Does this support the hypothesis that the population mean is 0? (Table value: 2.145)
Answers
Answer:
Step-by-step explanation:
The confidence interval formulas in the previous section are based on the Central Limit Theorem, the statement that for large samples X−− is normally distributed with mean μ and standard deviation σ/n−−√. When the population mean μ is estimated with a small sample (n < 30), the Central Limit Theorem does not apply. In order to proceed we assume that the numerical population from which the sample is taken has a normal distribution to begin with. If this condition is satisfied then when the population standard deviation σ is known the old formula x−−±zα∕2(σ/n−−√) can still be used to construct a 100(1−α)% confidence interval for μ.
If the population standard deviation is unknown and the sample size n is small then when we substitute the sample standard deviation s for σ the normal approximation is no longer valid. The solution is to use a different distribution, called Student’s t-distribution with n−1 degrees of freedom. Student’s t-distribution is very much like the standard normal distribution in that it is centered at 0 and has the same qualitative bell shape, but it has heavier tails than the standard normal distribution does, as indicated by Figure 7.5 "Student’s ", in which the curve (in brown) that meets the dashed vertical line at the lowest point is the t-distribution with two degrees of freedom, the next curve (in blue) is the t-distribution with five degrees of freedom, and the thin curve (in red) is the standard normal distribution. As also indicated by the figure, as the sample size n increases, Student’s t-distribution ever more closely resembles the standard normal distribution. Although there is a different t-distribution for every value of n, once the sample size is 30 or more it is typically acceptable to use the standard normal distribution instead, as we will always do in this text.