A survey of 77 commercial airline flights of under 2 hours resulted in a sample average late time for a flight of 2.48 minutes. The population standard deviation was 12 minutes. Construct a 95% confidence interval for the average time that a commercial flight of under 2 hours is late. What is the point estimate? What does the interval tell about whether the average flight is late?

#### Solution

Given that sample size $n = 77$, sample mean $\overline{X}= 2.48$ and population standard deviation $\sigma = 12$.

The point estimate of average late time for a flight is $\hat{\mu} = \overline{X} = 2.48$.

#### Step 1 Specify the confidence level $(1-\alpha)$

Confidence level is $1-\alpha = 0.95$. Thus, the level of significance is $\alpha = 0.05$.

#### Step 2 Given information

Given that sample size $n =77$, sample mean $\overline{X}=2.48$ and population standard deviation is $\sigma = 12$.

#### Step 3 Specify the formula

$100(1-\alpha)$\% confidence interval for the population mean $\mu$ is

` $$ \begin{aligned} \overline{X} - E \leq \mu \leq \overline{X} + E \end{aligned} $$ `

where `$E = Z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$`

, and `$Z_{\alpha/2}$`

is the $Z$ value providing an area of $\alpha/2$ in the upper tail of the standard normal probability distribution.

#### Step 4 Determine the critical value

The critical value of $Z$ for given level of significance is $Z_{\alpha/2}$.

Thus `$Z_{\alpha/2} = Z_{0.025} = 1.96$`

.

#### Step 5 Compute the margin of error

The margin of error for mean is

` $$ \begin{aligned} E & = Z_{\alpha/2} \frac{\sigma}{\sqrt{n}}\\ & = 1.96 \frac{12}{\sqrt{77}} \\ & = 2.68. \end{aligned} $$ `

#### Step 6 Determine the confidence interval

$95$ % confidence interval estimate for population mean is

` $$ \begin{aligned} \overline{X} - E & \leq \mu \leq \overline{X} + E\\ 2.48 - 2.68 & \leq \mu \leq 2.48 + 2.68\\ -0.2 & \leq \mu \leq 5.16. \end{aligned} $$ `

Thus, $95$% confidence interval for the average time that a commercial flight of under 2 hours late is $(-0.2,5.16)$.

Since the confidence interval includes 0, it indicate that the average flight is late.