Question

a psychiatrist is interested in finding a 95% confidence interval for the tics per hour exhibited by children with tourette syndrome. the data below show the tics in an observed hour for 15 randomly selected children with tourette syndrome. round answers to 3 decimal places where possible.
7 11 2 1 2 0 11 8 11 1 0 3 10 7 11
a. to compute the confidence interval use a t distribution.
b. with 95% confidence the population mean number of tics per hour that children with tourette syndrome exhibit is between and.
c. if many groups of 15 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 ($\bar{x}$)

$$\bar{x}=\frac{7 + 11+2 + 1+2 + 0+11 + 8+11 + 1+0 + 3+10 + 7+11}{15}=\frac{85}{15}\approx5.667$$

Step2: Calculate sample standard deviation ($s$)

First, calculate the deviations from the mean:

$$
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Sum of squared deviations:

$$1.777+28.444 + 13.444+21.777+13.444+32.111+28.444+5.444+28.444+21.777+32.111+7.111+18.777+1.777+28.444=303.333$$

Sample variance \(s^{2}=\frac{303.333}{15 - 1}=\frac{303.333}{14}\approx21.667\)
Sample standard deviation \(s=\sqrt{21.667}\approx4.655\)

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

Degrees of freedom \(df=n - 1=15-1 = 14\), \(\alpha=1 - 0.95=0.05\), \(\alpha/2=0.025\)
From \(t\)-distribution table, \(t_{0.025,14}=2.145\)

Step4: Calculate margin of error ($E$)

$$E=t_{\alpha/2}\frac{s}{\sqrt{n}}=2.145\times\frac{4.655}{\sqrt{15}}\approx2.145\times1.202\approx2.578$$

Step5: Calculate confidence interval

Lower limit \(\bar{x}-E=5.667-2.578 = 3.089\)
Upper limit \(\bar{x}+E=5.667 + 2.578=8.245\)

Answer:

b. With 95% confidence the population mean number of tics per hour that children with Tourette syndrome exhibit is between \(3.089\) and \(8.245\)
c. If many groups of 15 randomly selected children with Tourette syndrome are observed, then a different confidence interval would be produced from each group. About \(95\) percent of these confidence intervals will contain the true population mean number of tics per hour and about \(5\) percent will not contain the true population mean number of tics per hour.