Question

The mean age of graduate students at a University is at most $31$years with a standard deviation of two years. A random sample of $15$graduate students is taken. The sample mean is $32$years and the sample standard deviation is three years. Are the data significant at the $1\%$ level? The p-value is $0.0264$. State the null and alternative hypotheses and interpret the p-value

Short answer

The data are not significant at $1\%$level.

The null hypothesis is,

$H_0:\mu \le 31$,

and the alternate hypothesis is,

$H_a:\mu >31$

Step-by-step solution

Introduction 

The null hypothesis is a statistical theory that states that no statistical link or significance exists between two sets of observed data and measured events in a given single observed variable.

Explanation

The standard variance of a normal distribution is two years. We'd like to double-check a claim that graduate students at a university are on average$31$years old. As a result, the null hypothesis is true.

$H_0:\mu \le 31$

as well as the alternative hypothesis is,

$H_a:\mu >31$

Explanation

The $p$- value is $0.0264$, and the $\alpha =1\%=0.01$. Therefore,

$p$-value$=0.0264>0.01=\alpha$

As a result, we are unable to dismiss the entire theory. At the $1\%$level, there is enough evidence to establish that the average age of graduate students at a university is at most $31$years

We conclude that the findings are not significant at the $1\%$level because the sample mean is $32$years old.