If you take 40 samples from a concrete batch and find a samp

If you take 40 samples from a concrete batch and find a sample mean of 30.4 MPa and a standard deviation of 4 MPa, what is your 95th% confidence interval for the average sample mean?

If you need to ensure that the concrete batch you are using has an average compressive strength of at least 30 MPa, state the null and alternative hypotheses, in both sentence and equation form.

Conduct a statistical test to evaluate the null hypothesis, and state your conclusion.

Solution

a)
Confidence Interval
CI = x ± t a/2 * (sd/ Sqrt(n))
Where,
x = Mean
sd = Standard Deviation
a = 1 - (Confidence Level/100)
ta/2 = t-table value
CI = Confidence Interval
Mean(x)=30.4
Standard deviation( sd )=4
Sample Size(n)=40
Confidence Interval = [ 30.4 ± t a/2 ( 4/ Sqrt ( 40) ) ]
= [ 30.4 - 2.0227 * (0.63246) , 30.4 + 2.0227 * (0.63246) ]
= [ 29.12073,31.67927 ]
b)
Set Up Hypothesis
Null, H0: U<30
Alternate, H1: U>30
Test Statistic
Population Mean(U)=30
Sample X(Mean)=30.4
Standard Deviation(S.D)=4
Number (n)=40
we use Test Statistic (t) = x-U/(s.d/Sqrt(n))
to =30.4-30/(4/Sqrt(39))
to =0.632
| to | =0.632
Critical Value
The Value of |t | with n-1 = 39 d.f is 1.685
We got |to| =0.632 & | t | =1.685
Make Decision
Hence Value of |to | < | t | and Here we Do not Reject Ho
P-Value :Right Tail - Ha : ( P > 0.6325 ) = 0.26539
Hence Value of P0.05 < 0.26539,Here We Do not Reject Ho