Points: 0 of 1
Construct the confidence interval for the population mean $\mu$.
$$c=0.95,\overline{x}=7.8,\sigma =0.5,\text{ and }n=55$$

A $95\mathrm{\%}$ confidence interval for $\mu$ is $(◻,◻$. (Round to two decimal places as needed.)

Step 1 ：Given values are: confidence level $c=0.95$, sample mean $\overline{x}=7.8$, standard deviation $\sigma =0.5$, and sample size $n=55$.

Step 2 ：The z-score for a 95% confidence level is $z=1.96$.

Step 3 ：Calculate the margin of error using the formula: $margin\mathrm{\_}of\mathrm{\_}error=z\times \frac{\sigma }{\sqrt{n}}$. Substituting the given values, we get $margin\mathrm{\_}of\mathrm{\_}error=0.13214317304279546$.

Step 4 ：Calculate the confidence interval using the formula: $lower\mathrm{\_}limit=\overline{x}-margin\mathrm{\_}of\mathrm{\_}error$ and $upper\mathrm{\_}limit=\overline{x}+margin\mathrm{\_}of\mathrm{\_}error$. Substituting the given values, we get $lower\mathrm{\_}limit=7.6678568269572045$ and $upper\mathrm{\_}limit=7.932143173042795$.

Step 5 ：Round the lower and upper limits to two decimal places. The final 95% confidence interval for $\mu$ is $\boxed{{\displaystyle (7.67,7.93)}}$.