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.
\[ \bar{x}=\frac{\sum x_i}{n} \] \[ \bar{x}=\frac{150}{25}=6 \]

Step 2: Find the sample variance.
\[ s^2=\frac{\sum x_i^2-\frac{(\sum x_i)^2}{n}}{n-1} \] \[ s^2=\frac{1116-\frac{150^2}{25}}{24} \] \[ s^2=\frac{1116-900}{24} \] \[ s^2=\frac{216}{24}=9 \]
Therefore,
\[ s=3 \]

Step 3: Use the \(t\)-statistic formula.
\[ t=\frac{\bar{x}-\mu_0}{s/\sqrt{n}} \]
Here,
\[ \bar{x}=6,\quad \mu_0=4.8,\quad s=3,\quad n=25 \]

Step 4: Substitute the values.
\[ t=\frac{6-4.8}{3/\sqrt{25}} \] \[ t=\frac{1.2}{3/5} \] \[ t=\frac{1.2}{0.6} \] \[ t=2 \]

Step 5: Final conclusion.
Hence, the calculated value of the \(t\)-statistic is
\[ \boxed{2} \]