A sample of size 100 is chosen from a normally distributed mean,(µ) and standard deviation 60. A researcher has made the statement as follow to conclude that value of µ:
" reject H0:µ= 80 and accept H1: µ < 80. if x̅ < 68 "
a) if the significant level used is ∝%, find the value of ∝.
Z = (score-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.
what is the score.. can u pls explain...
To find the value of α (the significance level) in this scenario, we need to refer to the probability distribution of a normally distributed sample mean.
Given that the sample is chosen from a normally distributed population with a known standard deviation of 60, we can use the standard normal distribution to find the critical value associated with the rejection region.
Since the researcher is rejecting the null hypothesis H0: µ = 80 and accepting the alternative hypothesis H1: µ < 80 when x̅ < 68, we can interpret this as a one-tailed test. In this case, the critical value corresponds to the z-score associated with the chosen significance level α.
To find the critical z-score, we can use a z-table or a calculator. Since the researcher wants to reject the null hypothesis for values of x̅ less than 68 (µ < 80), we are interested in the left-tail area.
The critical value z α can be calculated as:
z α = Z(α)
Substituting Z(α) with the desired left-tail area in the z-table, we can find the corresponding z-score. In this case, Z(α) = α, as the z-table provides the area up to a given z-score.
Once we find the z α value, we can use the standard normal distribution table to find the corresponding probability.
Finally, the researcher's stated significance level ∝ is equal to 1 - the probability associated with the critical value z α.
Thus, to find the value of ∝, we need to calculate the left-tail area up to the critical z-score and subtract it from 1.
Note: The specific value of ∝ (alpha) is not mentioned in the question, so we cannot provide an exact numerical answer. However, by following the steps outlined above, you can find the value of ∝ based on the desired significance level.