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Sagot :
ANSWER:
Lower limit: 0.77
Upper limit: 0.84
STEP-BY-STEP EXPLANATION:
Given:
x = 303
n = 375
We calculate the value of the proportion in the following way:
[tex]\begin{gathered} p=\frac{x}{n}=\frac{303}{375} \\ \\ p=0.808 \end{gathered}[/tex]For a 90% confidence interval we have the following:
[tex]\begin{gathered} \alpha=100\%-90\%=10\%=0.1 \\ \\ \alpha\text{/2}=0.1=0.05 \\ \\ \text{ For the normal table this corresponds to:} \\ \\ Z_{\alpha\text{/2}}=1.645 \end{gathered}[/tex]We calculate the limits of the 90% confidence interval using the following formula:
[tex]\begin{gathered} \text{ Lower limit: }p-Z_{\alpha\text{/2}}\cdot\sqrt{\frac{p\cdot(1-p)}{n}}=\:0.808-1.645\cdot\sqrt{\frac{0.808\cdot\left(1-0.808\right)}{375}}\:=0.77 \\ \\ \:\text{Upper limit: }p-Z_{\alpha\text{/2}}\cdot\sqrt{\frac{p\cdot\left(1-p\right)}{n}}\:=0.808+1.645\cdot\sqrt{\frac{0.808\cdot\left(1-0.808\right)}{375}}=0.84 \end{gathered}[/tex]
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