Question

consider the hypotheses below
$h_0: \mu = 50$
$h_1: \mu \
eq 50$
given that $\bar{x}=57$, $s = 8$, $n = 20$, and $\alpha=0.01$, answer the questions below
a. what conclusion should be drawn?
b. use technology to determine the p - value for this test
the critical value(s) is(are)
(round to three decimal places as needed. write the numbers in increasing order. use a comma to separate answers as needed )
determine the test statistic, $t_0$
$t_0=$
(round to two decimal places as needed.)
what conclusion should be drawn? choose the correct answer below
○ a. reject the null hypothesis. the data provide sufficient evidence to conclude that the mean differs from $\mu = 50$
○ b. do not reject the null hypothesis. the data provide sufficient evidence to conclude that the mean differs from $\mu = 50$
○ c. reject the null hypothesis. the data do not provide sufficient evidence to conclude that the mean differs from $\mu = 50$

Explanation:

Step1: Calculate degrees of freedom

Degrees of freedom \(df=n - 1=20-1 = 19\)

Step2: Find critical values

For a two - tailed test with \(\alpha = 0.01\) and \(df = 19\), using the t - distribution table or technology, the critical values are \(t_{\alpha/2,df}=t_{0.005,19}\approx\pm 2.861\)

Step3: Calculate the test statistic

The formula for the t - test statistic is \(t_{0}=\frac{\bar{x}-\mu}{s/\sqrt{n}}\)
Substitute \(\bar{x} = 57\), \(\mu = 50\), \(s = 8\), \(n = 20\)

$$
LATEXBLOCK0
$$

Step4: Make a decision

Since \(|t_{0}|=3.91>2.861\) (the critical value), we reject the null hypothesis.

Answer:

• Critical values: \(- 2.861,2.861\)
• Test statistic \(t_{0}\approx3.91\)
• A. Reject the null hypothesis. The data provide sufficient evidence to conclude that the mean differs from \(\mu = 50\)