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Answer:
We conclude that the true average percentage of Silicon Dioxide is smaller than 5.5.
Step-by-step explanation:
We are given that the desired percentage of Silicon Dioxide (SiO2) in a certain type of aluminous cement is 5.5.
16 independently obtained samples are analyzed and a sample mean of 5.25 was obtained. Suppose that the percentage of SiO2 in a sample is normally distributed with a sigma of 0.3.
Let = <u><em>true average percentage of Silicon Dioxide.</em></u>
So, Null Hypothesis, :
5.5 {means that the true average is greater than or equal to 5.5}
Alternate Hypothesis, :
< 5.5 {means that the true average is smaller than 5.5}
The test statistics that would be used here <u>One-sample z test statistics</u> as we know about the population standard deviation;
T.S. = ~ N(0,1)
where, = sample mean percentage of Silicon Dioxide = 5.25
σ = population standard deviation = 0.3
n = sample size = 16
So, <em><u>the test statistics</u></em> =
= -3.33
The value of z test statistics is -3.33.
<u>Now, the P-value of the test statistics is given by;</u>
P-value = P(Z < -3.33) = 1 - P(Z 3.33)
= 1 - 0.9996 = <u>0.0004</u>
Since, the P-value of the test statistics is less than the level of significance as 0.0004 < 0.01, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region due to which <u>we reject our null hypothesis</u>.
Therefore, we conclude that the true average percentage of Silicon Dioxide is smaller than 5.5.