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The following random sample was selected from a normal distribution:

{:[9,4,9,7,2,20,14,10,20,14]:}
(a) Construct a
90% confidence interval for the population mean
mu.
◻
<= mu <=
◻
(b) Construct a
95% confidence interval for the population mean
mu.
◻

<= mu <=

◻

The following random sample was selected from a normal distribution: $\newline$ $\begin{array}{llllllllll} 9 & 4 & 9 & 7 & 2 & 20 & 14 & 10 & 20 & 14 \end{array}$ $\newline$ (a) Construct a $90 \%$ confidence interval for the population mean $\mu$ . $\square$ $\leq \mu \leq$ $\square$ $\newline$ (b) Construct a $95 \%$ confidence interval for the population mean $\mu$ . $\square$ $\newline$ $\leq \mu \leq$ $\newline$ $\square$

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Interpret confidence intervals for population means

Full solution

Q. The following random sample was selected from a normal distribution:

\newline

\begin{array}{llllllllll} 9 & 4 & 9 & 7 & 2 & 20 & 14 & 10 & 20 & 14 \end{array}

\newline

(a) Construct a

90 \%

confidence interval for the population mean

\mu

.

\square

\leq \mu \leq

\square

\newline

(b) Construct a

95 \%

confidence interval for the population mean

\mu

.

\square

\newline

\leq \mu \leq

\newline

\square

Calculate Mean and Standard Deviation: Calculate the sample mean ( $\bar{x}$ ) and sample standard deviation ( $s$ ) from the given data: \{ $9$ , $4$ , $9$ , $7$ , $2$ , $20$ , $14$ , $10$ , $20$ , $14$ \}. $\newline$ - Sample mean ( $\bar{x}$ ) = $(9 + 4 + 9 + 7 + 2 + 20 + 14 + 10 + 20 + 14) / 10 = 109 / 10 = 10.9$ $\newline$ - Sample standard deviation ( $s$ ) = $\sqrt{[(9-10.9)^2 + (4-10.9)^2 + (9-10.9)^2 + (7-10.9)^2 + (2-10.9)^2 + (20-10.9)^2 + (14-10.9)^2 + (10-10.9)^2 + (20-10.9)^2 + (14-10.9)^2] / (10-1)}$ $\newline$ = $\sqrt{[3.61 + 47.61 + 3.61 + 15.21 + 79.21 + 82.81 + 9.61 + 0.81 + 82.81 + 9.61] / 9}$ $\newline$ = $\sqrt{[334.3] / 9}$ $\newline$ = $\sqrt{37.144}$ $\newline$ = $6$ . $095$
Calculate t-values for Confidence Levels: Calculate the t-values for $90\%$ and $95\%$ confidence levels for a sample size of $10$ (degrees of freedom = $9$ ). $\newline$ - t-value for $90\%$ confidence $(t_{0.05}, 9) = 1.833$ $\newline$ - t-value for $95\%$ confidence $(t_{0.025}, 9) = 2.262$
Construct $90$ % Confidence Interval: Construct the $90$ % confidence interval for the population mean $\mu$ . $\newline$ - Margin of error = $t$ -value $\times (s / \sqrt{n})$ $\newline$ - Margin of error = $1.833 \times (6.095 / \sqrt{10})$ $\newline$ - Margin of error = $1.833 \times 1.928$ $\newline$ - Margin of error = $3.534$ $\newline$ - Confidence interval = $\bar{x} \pm$ Margin of error $\newline$ - Confidence interval = $10.9 \pm 3.534$ $\newline$ - Confidence interval = $(7.366, 14.434)$
Construct $95$ % Confidence Interval: Construct the $95$ % confidence interval for the population mean $\mu$ . $\newline$ - Margin of error = $t$ -value $\times$ $\frac{s}{\sqrt{n}}$ $\newline$ - Margin of error = $2.262 \times \frac{6.095}{\sqrt{10}}$ $\newline$ - Margin of error = $2.262 \times 1.928$ $\newline$ - Margin of error = $4.362$ $\newline$ - Confidence interval = $\bar{x} \pm$ Margin of error $\newline$ - Confidence interval = $10.9 \pm 4.362$ $\newline$ - Confidence interval = $(6.538, 15.262)$

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