91影视

Refer to Exercise 9 for the summary statistics for randomly selected weights of newborn girls.

Here,

\(n\, = \,205,\bar x\, = 30.4\,{\rm{hg}}\).

The 95% confidence level with the known value of \(\sigma = 7.1\,\,{\rm{hg}}\) .

A confidence interval is an estimate of the interval that may contain the true value of a population parameter. It is also known as an interval estimate.

The general formula for the confidence interval estimate of mean for the known\(\sigma \)is as follows.

\({\rm{Confidence}}\,\,{\rm{interval}} = \left( {\bar x - {\rm{E}},\bar x + {\rm{E}}} \right)\,\,\,\,\,\,\,\,...\left( 1 \right)\)\(\)

Here, E is the margin of error, which is calculated as follows.

\({\rm{E}} = {z_{\frac{\alpha }{2}}} \times \frac{\sigma }{{\sqrt {\rm{n}} }}\)

For a normally distributed population with randomly selected observations, the following are true.

If\(\sigma \)is known, the normal distribution is suitable to find the confidence interval.

If\(\sigma \)is unknown, the student鈥檚 t-distribution is suitable to find the confidence interval.

In this case,\(\sigma \)is known, and\({\rm{n}} = 205\),which is greater than 30.

Thus, normal distribution applies.

\({z_{\frac{\alpha }{2}}}\)is a z score that separates an area of\(\frac{\alpha }{2}\)in the right tail of the standard normal distribution.

The confidence level 95% corresponds to\(\alpha = 0.05\,\,\,\,{\rm{and}}\,\,\,\,\frac{\alpha }{2} = 0.025\).

The value\({z_{\frac{\alpha }{2}}}\)has the cumulative area\(1 - \frac{\alpha }{2}\)to its left. .

Mathematically,

\(\begin{array}{c}P\left( {z < {z_{\frac{\alpha }{2}}}} \right) = 1 - \frac{\alpha }{2}\\ = 0.975\end{array}\)

From the standard normal table, the area of 0.975 is observed corresponding to the row value 1.9 and column value 0.06, which implies that\({z_{\frac{\alpha }{2}}}\)is 1.96.

The margin error is calculated as follows.

\(\begin{array}{c}{\rm{E}} = {z_{\frac{\alpha }{2}}} \times \frac{\sigma }{{\sqrt {\rm{n}} }}\\ = 1.96 \times \frac{{7.1}}{{\sqrt {205} }}\\ = 0.9719\end{array}\).

The confidence interval is obtained by substituting the value of the margin of error in equation (1), as follows.

\(\begin{array}{c}{\rm{Confidence}}\,\,{\rm{interval}} = \left( {\bar x - E,\bar x + E} \right)\\ = \left( {30.4 - 0.9719\,\,,\,\,30.4 + 0.9719} \right)\\ = \left( {29.4281\,,\,\,31.3719} \right)\end{array}\)

Thus, the 95% confidence interval for the estimate mean is \(29.4\,{\rm{hg}} < \mu < 31.4\,{\rm{hg}}\).