$3.$ In a test of hypothesis, the null hypothesis is that the population mean is equal to $45$ and the alternative hypothesis is that the population mean is greater than $45.$ The test is to be made at the $2\mathrm{.}5\%$ significance level. A sample of $81$ elements selected from this population produced a mean of $47\mathrm{.}3$ and a standard deviation of $4\mathrm{.}5.$

Step $1$: Define the null and alternative hypotheses

Step $2$: Calculate the test statistic

Step $3$: Set Rejection Region

Step $4$: Conclude