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# A manager is interested in determining if the population standard deviation has dropped below 134. Based on a sample of n=27 items selected randomly from the​ population, conduct the appropriate hypothesis test at a 0.05 significance level. The sample standard deviation is 126. Determine the Null and alternative hypothese

Accepted Solution

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Answer:H₀: σ² ≥ 17956H₁: σ² < 17956Step-by-step explanation:Hello!You are asked to test if the population standard deviation of a certain population. Now keep in mind that wherever you want to make a hypothesis test for a population parameter, you have to have a known distribution that includes this parameter. Since the population standard deviation is no parameter of any distribution, what you have to do is test the population variance. Any decision you make about the population variance can be extrapolated to the population standard deviation.To make a hypothesis test for the population variance, you need a variable with normal distribution and the statistic to use is a Chi-square statistic.The hypothesis is that the population standard deviation is less than 134, symbolically: σ < 134Translated in terms of the population variance: σ² < 17956H₀: σ² ≥ 17956H₁: σ² < 17956α: 0.05χ²=  (n-1)S²  ~χ²$$_{n-1}$$ σ²χ²=  (27-1)(126)²  = 22.988 17956The test is one tailed (left)χ²$$_{n-1;α}$$ =χ²$$_{26;0.05}$$ = 15.379If the calculated Chi-square value is ≤ than the critical value, you reject the null hypothesis.If the calculated Chi-square value is > than the critical value, you don't reject the null hypothesis.Since the value is greater than the critical value, you do not reject the null hypothesis. So at a 5% level, there is not enough evidence to reject the null hypothesis, this means the population variance is at least 17956. On the same level, you can conclude that the population standard deviation is at least 134.I've made the test so that you have an example of how to do it.I hope it helps!