Determine the rejection region in the problem involving hypothesis

[chwala]

Homework Statement

See attached

Relevant Equations

stats

Ms Solution

My approach is just the same as ms...

The thing to note is that at $0.975= Z_{[-1.96]}$

therefore we shall have,

$-1.96=\left[\dfrac{X-45.7}{\dfrac{5.6}{\sqrt 30}}\right]=\dfrac{X-45.7}{1.022415441}$

$-1.96×1.022415441=X-45.7$

$X=-2.003934+45.7=43.7$

Any insight is welcome.

 

[mighty2000]

Hello Ms Solution,

Thank you for sharing your approach with us. I agree with your calculations and conclusion that the value of X is 43.7. To provide further insight, I would like to explain the reasoning behind the use of the Z-score in this problem.

The Z-score is a measure of how many standard deviations a data point is above or below the mean. In this case, we are given the confidence level of 0.975, which corresponds to a Z-score of -1.96. This means that the data point (X) we are looking for is 1.96 standard deviations below the mean of 45.7.

To solve for X, we use the formula for the Z-score, which is (X - mean) / standard deviation. In this case, the mean is 45.7 and the standard deviation is 5.6 / √30. By substituting these values into the formula and solving for X, we get the same result as you did.

I hope this provides a better understanding of the use of Z-score in this problem. Thank you for sharing your approach and opening the discussion for further insights. Keep up the great work!