Susan is conducting a survey to understand the electric bills of households in her city. She finds that the average (mean) electric bill for the population is [tex]$98.75$[/tex] with a standard deviation of [tex]$10.45$[/tex]. With a sample size of [tex]$60$[/tex] households, Susan wants to find the interval in which she can be [tex]$68\%$[/tex] confident that the sample mean will lie.
To do this, Susan needs to calculate the confidence interval for the sample mean. The steps involved in calculating this are as follows:
1. Find the z-score corresponding to the confidence level: For a [tex]$68\%$[/tex] confidence level, the z-score is approximately [tex]$1$[/tex]. This is because [tex]$68\%$[/tex] is within [tex]$1$[/tex] standard deviation of the mean in a normal distribution.
2. Calculate the Standard Error (SE) of the sample mean: The formula for the standard error is
[tex]\[
\text{SE} = \frac{\sigma}{\sqrt{n}}
\][/tex]
where [tex]$\sigma$[/tex] is the population standard deviation and [tex]$n$[/tex] is the sample size.
Here,
[tex]\[
\text{SE} = \frac{10.45}{\sqrt{60}} \approx 1.3491
\][/tex]
3. Calculate the Margin of Error (MOE): The margin of error is computed using the z-score and the standard error:
[tex]\[
\text{MOE} = z \times \text{SE}
\][/tex]
With [tex]$z = 1$[/tex] and [tex]$\text{SE} \approx 1.3491$[/tex],
[tex]\[
\text{MOE} = 1 \times 1.3491 \approx 1.3491
\][/tex]
4. Calculate the Confidence Interval: The confidence interval is given by
[tex]\[
(\text{Population mean} - \text{MOE}, \text{Population mean} + \text{MOE})
\][/tex]
Thus,
[tex]\[
(98.75 - 1.3491, 98.75 + 1.3491) = (97.4009, 100.0991)
\][/tex]
Therefore, to complete the equation that Susan can use to find the interval in which she can be [tex]$68\%$[/tex] sure that the sample mean will lie, we can fill in the following values:
- Population mean ([tex]$\bar{x}$[/tex]): [tex]$98.75$[/tex]
- Standard deviation ([tex]$\sigma$[/tex]): [tex]$10.45$[/tex]
- Sample size ([tex]$n$[/tex]): [tex]$60$[/tex]
So the equation for the [tex]$68\%$[/tex] confidence interval is:
[tex]\[
(98.75 - 1.3491, 98.75 + 1.3491) = (97.4009, 100.0991)
\][/tex]
