Answer :
The tri-linear inequality with the required 80% confidence interval of 54.1 to 57.7 shows the genuine mean value by which the drug decreases the average patient's systolic blood pressure.
A confidence interval expresses the likelihood that a population parameter will fall between a range of values a predetermined percentage of the time. It is the estimate's mean plus or minus the estimate's range of values.
Given the confidence level is 80%, the significance level α=100-80=20% = 0.20. The sample mean [tex]\overline{x}[/tex] is 55.9, the sample standard deviation s is 7.4, and the sample size n is 29.
We are using t-distribution to population standard deviation first. For that first find the degree of freedom and critical value of t.
Degree of freedom,
df = n - 1
df = 29 - 1
df = 28.
The critical value of t calculated from the t-distribution table is,
[tex]\begin{aligned}t_{\text{critical}}&=t_{\alpha/2,df}\\&=t_{0.10,28}\\&=\pm1.3125 \end{aligned}[/tex]
Then, an 80% confidence interval is,
[tex]\begin{aligned}\mu&=\overline{x}\pm\frac{t\cdot s}{\sqrt{n}}\\&=55.9\pm\frac{1.3125\times7.4}{\sqrt{29}}\\&=55.9\pm1.80\end{aligned}[/tex]
Then, the interval is written as
55.9 - 1.80 < μ < 55.9 + 1.80
54.1 < μ < 57.7
The required interval is 54.1 < μ < 57.7.
To know more about confidence intervals:
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