Question

In a survey of 3467 adults aged 57 through 85 years, it was found that 83.7% of them used at least one prescription medication. Complete parts (a) through (c) below.a. How many of the 3467 subjects used at least one prescription medication?2902 (Round to the nearest Integer as needed.)b. Construct a 90% confidence interval estimate of the percentage of adults aged 57 through 85 years who use at least one prescription medication.%

Answer 1

We are given the following information.

Proportion = p = 83.7%

Sample size = n = 3467

Confidence level = 90%

b. Construct a 90% confidence interval estimate of the percentage of adults aged 57 through 85 years who use at least one prescription medication

The confidence interval is given by

`\begin{gathered} CI=(p\pm MoE) \\ CI=(p-MoE,\: p+MoE) \end{gathered}`

Where MoE is the margin of error and is given by

`MoE=z_{(\alpha)/(2)}\cdot\sqrt[]{(p(1-p))/(n)}`

Where

`z_{(\alpha)/(2)}=1-0.90=(0.10)/(2)=0.05`

From the normal distribution table, the value of z_0.05 corresponding to a 90% confidence interval is found to be 1.65

So, the margin of error is

`\begin{gathered} MoE=z_{(\alpha)/(2)}\cdot\sqrt[]{(p(1-p))/(n)} \\ MoE=1.65_{}\cdot\sqrt[]{(0.837(1-0.837))/(3467)} \\ MoE=1.65_{}\cdot\sqrt[]{(0.1364)/(3467)} \\ MoE=1.65_{}\cdot0.00627 \\ MoE=0.0103 \end{gathered}`

So, the confidence interval is

[tex]\begin{gathered} CI=(p-MoE,\: p+MoE) \\ CI=(0.837-0.0103,\: 0.837+0.0103) \\ CI=(0.8267,\: 0.8473) \\ CI=(82.7\%,\: 84.7\%) \\ or \\ 82.7\%

What does it mean?

It means that we are 90% confident that the percentage of adults aged 57 through 85 years who use at least one prescription medication lies within the interval of (82.7%, 84.7%)