Question

It is given that the variance of a population is \(4\), but its mean \((\mu)\) is unknown. A sample of size \(25\) is drawn randomly from this population to test the null hypothesis \(H_0:\mu=4.8\). The sample elements are \(x_1,x_2,\ldots,x_{25}\) such that \(\sum_{i=1}^{25}x_i=150\) and \(\sum_{i=1}^{25}x_i^2=1116\). Based on the above information, the calculated value of \(t\)-statistic is (in integer).

Hint:

For a one-sample \(t\)-statistic, first calculate the sample mean and sample standard deviation, then use \(t=\frac{\bar{x}-\mu_0}{s/\sqrt{n}}\).

Answer 1

Correct Answer: 2

Step 1: Find the sample mean.
\[ \bar{x}=\frac{\sum x_i}{n}=\frac{150}{25}=6 \]

Step 2: Find the sample variance.
\[ s^2=\frac{\sum x_i^2-\dfrac{(\sum x_i)^2}{n}}{n-1}=\frac{1116-900}{24}=\frac{216}{24}=9 \]
So $s=3$.

Step 3: Write the t statistic.
\[ t=\frac{\bar{x}-\mu_0}{s/\sqrt{n}} \]
with $\mu_0=4.8$ and $n=25$.

Step 4: Plug in.
\[ t=\frac{6-4.8}{3/5}=\frac{1.2}{0.6}=2 \]
\[ \boxed{2} \]