Answer:
(a) The value
= 72.3 is 1.21 standard deviations below the null value of <em>μ</em>.
(b) The mean drying time of paint is not less than 74.
Step-by-step explanation:
The hypothesis for the single mean test is:
<em>H₀</em>: <em>μ</em> = 74 vs. <em>Hₐ</em>: μ < 74.
The information provided is:
<em>σ</em> = 7
<em>n</em> = 25
As the population standard deviation is known, we will use a <em>z</em>-test for single mean.
(a)
The <em>z</em>-score is a Normal distribution with mean 0 and variance 1. It is also defined as the number of standard deviations a raw score is from the mean.
The <em>z</em>-score for sample mean is given by:
![z=\frac{\bar x-\mu}{\sigma/\sqrt{n}}](https://tex.z-dn.net/?f=z%3D%5Cfrac%7B%5Cbar%20x-%5Cmu%7D%7B%5Csigma%2F%5Csqrt%7Bn%7D%7D)
If
= 72.3 compute the corresponding <em>z</em>-score as follows:
![z=\frac{\bar x-\mu}{\sigma/\sqrt{n}}](https://tex.z-dn.net/?f=z%3D%5Cfrac%7B%5Cbar%20x-%5Cmu%7D%7B%5Csigma%2F%5Csqrt%7Bn%7D%7D)
![=\frac{72.3-74}{7/\sqrt{25}}\\=-1.21](https://tex.z-dn.net/?f=%3D%5Cfrac%7B72.3-74%7D%7B7%2F%5Csqrt%7B25%7D%7D%5C%5C%3D-1.21)
Thus, the value
= 72.3 is 1.21 standard deviations below the null value of <em>μ</em>.
(b)
So the test statistic is, <em>z</em> = -1.21.
Compute the <em>p</em>-value of the test as follows:
![p-value=P(Z](https://tex.z-dn.net/?f=p-value%3DP%28Z%3Cz%29%5C%5C%3DP%28Z%3C-1.21%29%5C%5C%3D0.1131)
*Use a <em>z</em>-table.
The decision rule says to, reject the null hypothesis if the <em>p</em>-value is less than the significance level <em>α</em> = 0.01 and vice-versa.
<em>p</em>-value = 0.1131 < <em>α</em> = 0.01
The null hypothesis was failed to be rejected.
Thus, the mean drying time of paint is not less than 74.