Question

suppose you are a researcher in a hospital. you are experimenting with a new sedative. you collect data from a random sample of 9 patients. the period of effectiveness of the sedative for each patient (in hours) is as follows. click this link download csv to download the data. a) what is a point estimate for the population mean length of time? (round answer to 4 decimal places) b) what must be true in order to construct a confidence interval in this situation? c) construct a 90% confidence interval for the population mean length of time. enter your answer as an open - interval (i.e., parentheses example (5.2314,8.1245)) round upper and lower bounds to 4 decimal places. d) what does it mean to be \90% confident\ in this problem? the confidence interval contains 90% of all sample times. 90% of all times will fall within this interval.

Explanation:

Step1: Calculate the sample mean ($\bar{x}$)

The formula for the sample mean is $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$.
Given data: $x = [2.6,2.6,3,2.1,2.6,2.8,2.1,2.7,3]$ and $n = 9$.
$\sum_{i=1}^{9}x_{i}=2.6 + 2.6+3+2.1+2.6+2.8+2.1+2.7+3=23.1$
$\bar{x}=\frac{23.1}{9}=2.5667$ (rounded to 4 decimal places)

Step2: Calculate the sample standard deviation ($s$)

The formula for the sample standard deviation is $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$.
First, calculate $(x_{i}-\bar{x})^{2}$ for each $x_{i}$:
$(2.6 - 2.5667)^{2}=0.0011$
$(2.6 - 2.5667)^{2}=0.0011$
$(3 - 2.5667)^{2}=0.1878$
$(2.1 - 2.5667)^{2}=0.2178$
$(2.6 - 2.5667)^{2}=0.0011$
$(2.8 - 2.5667)^{2}=0.0544$
$(2.1 - 2.5667)^{2}=0.2178$
$(2.7 - 2.5667)^{2}=0.0178$
$(3 - 2.5667)^{2}=0.1878$

$\sum_{i = 1}^{9}(x_{i}-\bar{x})^{2}=0.0011+0.0011 + 0.1878+0.2178+0.0011+0.0544+0.2178+0.0178+0.1878 = 0.8867$

$s=\sqrt{\frac{0.8867}{9 - 1}}=\sqrt{\frac{0.8867}{8}}=\sqrt{0.1108}=0.3330$ (rounded to 4 decimal places)

Step3: Find the t - value

For a 90% confidence interval and $n-1=9 - 1 = 8$ degrees of freedom, using a t - distribution table or calculator, $t_{\alpha/2}=1.8595$

Step4: Calculate the margin of error ($E$)

The formula for the margin of error is $E=t_{\alpha/2}\frac{s}{\sqrt{n}}$
$E = 1.8595\times\frac{0.3330}{\sqrt{9}}=1.8595\times\frac{0.3330}{3}=1.8595\times0.111 = 0.2064$ (rounded to 4 decimal places)

Step5: Calculate the confidence interval

The confidence interval is $(\bar{x}-E,\bar{x}+E)$
$\bar{x}-E=2.5667- 0.2064=2.3603$
$\bar{x}+E=2.5667+0.2064=2.7731$

Answer:

$(2.3603,2.7731)$