Question

You wish to test the following claim H_a at a significance level of alpha =0.05

H_o : mu = 74.1
H_a : mu > 74.1

You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n=53 with m=77.5 and a standard deviation of SD =6.9

What is the test statistic for this sample? (Report answer accurate to three decimal places.)
test statistic = __________________

What is the p-value for this sample? (Report answer accurate to four decimal places.)
p-value = __________________

Answer 1

To calculate the test statistic for the given sample, we can use the formula for the t-statistic since we don't know the population standard deviation and our sample size is less than 30 (but is acceptable as the sample size is larger than 30, making the t-distribution approach normal distribution):

\[
t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}
\]

where:
- \(\bar{x} = 77.5\) (sample mean),
- \(\mu_0 = 74.1\) (population mean under the null hypothesis),
- \(s = 6.9\) (sample standard deviation),
- \(n = 53\) (sample size).

First, we will calculate the standard error (SE):

\[
SE = \frac{s}{\sqrt{n}} = \frac{6.9}{\sqrt{53}}.
\]

Calculating \(\sqrt{53}\):

\[
\sqrt{53} \approx 7.2801,
\]

Thus:

\[
SE \approx \frac{6.9}{7.2801} \approx 0.9485.
\]

Now we can calculate the t-statistic:

\[
t = \frac{77.5 - 74.1}{0.9485} \approx \frac{3.4}{0.9485} \approx 3.5784.
\]

Finally, rounding to three decimal places:

\[
\text{test statistic} = 3.578.
\]

Next, we need to calculate the p-value for the one-tailed t-test. We will use the degrees of freedom, which can be calculated as:

\[
df = n - 1 = 53 - 1 = 52.
\]

To find the p-value, we can use a t-distribution table or software/calculator. The p-value for a t statistic of 3.578 with 52 degrees of freedom can be found using a t-table or a statistical software/package.

Using statistical software or a calculator, we determine the p-value corresponding to the test statistic of 3.578 with 52 degrees of freedom.

The p-value is approximately \(0.0005\) (using appropriate statistical software or calculators).

Thus, the value to four decimal places is:

\[
\text{p-value} \approx 0.0005.
\]

### Summary of the results:

- **Test statistic**: \(3.578\)
- **p-value**: \(0.0005\)