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Question
Suppose heights of dogs, in inches, in a city are normally distributed and have a known population standard deviation of 7 inches and an unknown population mean. A random sample of 15 dogs is taken and gives a sample mean of 34 inches. Find the confidence interval for the population mean with a
99% confidence level.

z_(0.10)

z_(0.05)

z_(0.025)

z_(0.01)

z_(0.005)

1.282
1.645
1.960
2.326
2.576

You may use a calculator or the common
z values above.

Round all calculations to three decimal places, if necessary.

Question $\newline$ Suppose heights of dogs, in inches, in a city are normally distributed and have a known population standard deviation of $7$ inches and an unknown population mean. A random sample of $15$ dogs is taken and gives a sample mean of $34$ inches. Find the confidence interval for the population mean with a $99 \%$ confidence level. $\newline$ \begin{tabular}{|c|c|c|c|c|} $\newline$ \hline $\mathbf{z}_{0.10}$ & $\mathbf{z}_{0.05}$ & $\mathbf{z}_{0.025}$ & $\mathbf{z}_{0.01}$ & $\mathbf{z}_{0.005}$ \\ $\newline$ \hline $1$ . $282$ & $1$ . $645$ & $1$ . $960$ & $2$ . $326$ & $2$ . $576$ \\ $\newline$ \hline $\newline$ \end{tabular} $\newline$ You may use a calculator or the common $z$ values above. $\newline$ - Round all calculations to three decimal places, if necessary.

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Find values of normal variables

Full solution

Q. Question

\newline

Suppose heights of dogs, in inches, in a city are normally distributed and have a known population standard deviation of

7

inches and an unknown population mean. A random sample of

15

dogs is taken and gives a sample mean of

34

inches. Find the confidence interval for the population mean with a

99 \%

confidence level.

\newline

\begin{tabular}{|c|c|c|c|c|}

\newline

\hline

\mathbf{z}_{0.10}

&

\mathbf{z}_{0.05}

&

\mathbf{z}_{0.025}

&

\mathbf{z}_{0.01}

&

\mathbf{z}_{0.005}

\\

\newline

\hline

1

.

282

&

1

.

645

&

1

.

960

&

2

.

326

&

2

.

576

\\

\newline

\hline

\newline

\end{tabular}

\newline

You may use a calculator or the common

z

values above.

\newline

- Round all calculations to three decimal places, if necessary.

Identify Values: Identify the necessary values for calculating the confidence interval. $\newline$ We have: $\newline$ - The sample mean ( $\bar{x}$ ) = $34$ inches $\newline$ - The population standard deviation ( $\sigma$ ) = $7$ inches $\newline$ - The sample size (n) = $15$ $\newline$ - The confidence level = $99$ %
Find Z-Value: Find the appropriate z-value for the $99$ % confidence level. $\newline$ Since the confidence level is $99$ %, we need to find the z-value that corresponds to the remaining $1$ % of the area under the normal distribution curve. This $1$ % is split equally on both tails of the distribution, so we look for the z-value that corresponds to $0.5\%$ in one tail. $\newline$ From the given z-values, we use $z_{0.005}$ which is $2.576$ .
Calculate Margin of Error: Calculate the margin of error (E) using the z-value. $\newline$ The margin of error (E) is calculated using the formula: $\newline$ E = z * ( $\sigma$ / √n) $\newline$ Plugging in the values we have: $\newline$ E = $2$ . $576$ * ( $7$ / √ $15$ )
Perform Calculation: Perform the calculation for the margin of error. $\newline$ $E = 2.576 \times \left(\frac{7}{\sqrt{15}}\right)$ $\newline$ $E = 2.576 \times \left(\frac{7}{3.873}\right)$ $\newline$ $E = 2.576 \times 1.806$ $\newline$ $E \approx 4.654$
Calculate Confidence Interval: Calculate the confidence interval using the sample mean and the margin of error. $\newline$ The confidence interval is given by ( $\bar{x}$ - E, $\bar{x}$ + E). $\newline$ Lower limit = $\bar{x}$ - E = $34$ - $4$ . $654$ ≈ $29$ . $346$ $\newline$ Upper limit = $\bar{x}$ + E = $34$ + $4$ . $654$ ≈ $38$ . $654$
Round Interval: Round the confidence interval to three decimal places. $\newline$ Lower limit $\approx 29.346$ (rounded to three decimal places) $\newline$ Upper limit $\approx 38.654$ (rounded to three decimal places)

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\newline

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\newline

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\newline

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\newline

1

\newline

g(c)=-3

for at least one

c

-1

4

\newline

g(c)=3

for at least one

c

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1

\newline

g(c)=-3

for at least one

c

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1

\newline

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