Question

a psychiatrist is interested in finding a 98% confidence interval for the tics per hour exhibited by children with tourette syndrome. the data below show the tics in an observed hour for 11 randomly selected children with tourette syndrome. round answers to 3 decimal places where possible.
7 0 6 2 8 6 12 12 3 5 2
a. to compute the confidence interval use a distribution.
b. with 98% confidence the population mean number of tics per hour that children with tourette syndrome exhibit is between and.
c. if many groups of 11 randomly selected children with tourette syndrome are observed, then a different confidence interval would be produced from each group. about percent of these confidence intervals will contain the true population mean number of tics per hour and about percent will not contain the true population mean number of tics per hour.

Explanation:

Step1: Calculate sample mean $\bar{x}$

$$\bar{x}=\frac{7 + 0+6 + 2+8 + 6+12 + 12+3 + 5+2}{11}=\frac{63}{11}\approx5.727$$

Step2: Calculate sample standard deviation $s$

First, find the deviations from the mean:
\((7 - 5.727)=1.273\), \((0 - 5.727)=- 5.727\), \((6 - 5.727)=0.273\), \((2 - 5.727)=-3.727\), \((8 - 5.727)=2.273\), \((6 - 5.727)=0.273\), \((12 - 5.727)=6.273\), \((12 - 5.727)=6.273\), \((3 - 5.727)=-2.727\), \((5 - 5.727)=-0.727\), \((2 - 5.727)=-3.727\)

Then, calculate the sum of squared deviations:
\(1.273^{2}+(-5.727)^{2}+0.273^{2}+(-3.727)^{2}+2.273^{2}+0.273^{2}+6.273^{2}+6.273^{2}+(-2.727)^{2}+(-0.727)^{2}+(-3.727)^{2}\)
\(=1.621 + 32.799+0.074+13.891+5.167+0.074+39.351+39.351+7.437+0.529+13.891\)
\(=153.184\)

Sample variance \(s^{2}=\frac{153.184}{11 - 1}=\frac{153.184}{10}=15.3184\)

Sample standard deviation \(s=\sqrt{15.3184}\approx3.914\)

Step3: Find the critical value \(t_{\alpha/2}\)

For a \(98\%\) confidence interval, \(\alpha=1 - 0.98 = 0.02\), \(\alpha/2=0.01\), and degrees of freedom \(df=n - 1=11 - 1 = 10\)

Using the \(t -\)distribution table or a calculator, \(t_{0.01,10}=2.764\)

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

\(E=t_{\alpha/2}\frac{s}{\sqrt{n}}=2.764\times\frac{3.914}{\sqrt{11}}\)
\(=2.764\times\frac{3.914}{3.317}\approx2.764\times1.18\approx3.262\)

Step5: Calculate the confidence interval

Lower bound: \(\bar{x}-E=5.727-3.262 = 2.465\)
Upper bound: \(\bar{x}+E=5.727+3.262=8.989\)

Answer:

b. With \(98\%\) confidence the population mean number of tics per hour that children with Tourette syndrome exhibit is between \(2.465\) and \(8.989\)
c. About \(98\) percent of these confidence intervals will contain the true population mean number of tics per hour and about \(2\) percent will not contain the true population mean number of tics per hour.