The best point estimate for the population mean μ \mu μ is given by the sample mean xˉ \bar{x} xˉ: Point estimate=xˉ=87.0 \text{Point estimate} = \bar{x} = 87.0 Point estimate=xˉ=87.0
To calculate the margin of error, we use the formula: Margin of Error=Z⋅sn \text{Margin of Error} = Z \cdot \frac{s}{\sqrt{n}} Margin of Error=Z⋅ns where Z Z Z is the Z-score corresponding to the 95% confidence level, s s s is the sample standard deviation, and n n n is the sample size. For a 95% confidence level, Z=1.96 Z = 1.96 Z=1.96.
Substituting the values: Margin of Error=1.96⋅7.242≈2.18 \text{Margin of Error} = 1.96 \cdot \frac{7.2}{\sqrt{42}} \approx 2.18 Margin of Error=1.96⋅427.2≈2.18
The confidence interval for the mean is calculated as: xˉ±Margin of Error \bar{x} \pm \text{Margin of Error} xˉ±Margin of Error Thus, we have: 87.0±2.18 87.0 \pm 2.18 87.0±2.18 Calculating the lower and upper bounds: Lower Bound=87.0−2.18=84.82 \text{Lower Bound} = 87.0 - 2.18 = 84.82 Lower Bound=87.0−2.18=84.82 Upper Bound=87.0+2.18=89.18 \text{Upper Bound} = 87.0 + 2.18 = 89.18 Upper Bound=87.0+2.18=89.18 Therefore, the 95% confidence interval is: (84.82,89.18) (84.82, 89.18) (84.82,89.18)
Point estimate=87.0, Margin of error=2.18, Confidence interval=(84.82,89.18) \boxed{\text{Point estimate} = 87.0, \text{ Margin of error} = 2.18, \text{ Confidence interval} = (84.82, 89.18)} Point estimate=87.0, Margin of error=2.18, Confidence interval=(84.82,89.18)
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