Question

consider the following hypotheses and sample data, and then complete parts (a) and (b) below using \\( \alpha = 0.10 \\).
\\( h _ { 0 } : \mu = 17 \\)
\\( h _ { 1 } : \mu > 17 \\)
20 20 13 18 22 20 15 23 19 14
the critical value(s) is(are) \\( \square \\).
(round to three decimal places as needed. use a comma to separate answers as needed.)
determine the test statistic, \\( t _ { 0 } \\).
\\( t _ { 0 } = \square \\)
(round to two decimal places as needed.)
what conclusion should be drawn?
a. do not reject \\( h _ { 0 } \\). there is not sufficient evidence to conclude that \\( \mu > 17 \\).
b. reject \\( h _ { 0 } \\). there is sufficient evidence to conclude that \\( \mu > 17 \\).
c. reject \\( h _ { 0 } \\). there is not sufficient evidence to conclude that \\( \mu > 17 \\).
d. do not reject \\( h _ { 0 } \\). there is sufficient evidence to conclude that \\( \mu > 17 \\).
b) use technology to determine the p - value for this test.
\\( p - \text { value } = \square \\)
(round to three decimal places as needed)

Explanation:

Step1: Calculate sample mean and standard deviation

Sample data: \(x = [20, 20, 13, 18, 22, 20, 15, 23, 19, 14]\)
Sample size \(n = 10\)
Sample mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}=\frac{20 + 20+13 + 18+22 + 20+15 + 23+19 + 14}{10}=\frac{184}{10}=18.4\)
Sample standard deviation \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}\)
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=(20 - 18.4)^{2}+(20 - 18.4)^{2}+(13 - 18.4)^{2}+(18 - 18.4)^{2}+(22 - 18.4)^{2}+(20 - 18.4)^{2}+(15 - 18.4)^{2}+(23 - 18.4)^{2}+(19 - 18.4)^{2}+(14 - 18.4)^{2}\)
\(=2.56+2.56 + 29.16+0.16+12.96+2.56+11.56+21.16+0.36+19.36=102.4\)
\(s=\sqrt{\frac{102.4}{9}}\approx3.37\)

Step2: Find critical value

Degrees of freedom \(df=n - 1=10 - 1 = 9\), right - tailed test with \(\alpha = 0.10\)
Using t - distribution table or calculator, critical value \(t_{\alpha,df}=t_{0.10,9}=1.383\)

Step3: Calculate test statistic

Test statistic \(t_{0}=\frac{\bar{x}-\mu_{0}}{s/\sqrt{n}}\)
\(\mu_{0}=17\), \(\bar{x}=18.4\), \(s = 3.37\), \(n = 10\)
\(t_{0}=\frac{18.4 - 17}{3.37/\sqrt{10}}\approx\frac{1.4}{1.07}\approx1.31\)

Step4: Determine conclusion

Since \(t_{0}=1.3117\)

Step5: Calculate p - value

Using technology (e.g., t - test function in statistical software or calculator with \(t = 1.31\), \(df=9\), right - tailed)
\(p-\text{value}\approx0.111\)

Answer:

Critical value: \(1.383\)
\(t_{0}=1.31\)
Conclusion: A. Do not reject \(H_{0}\). There is not sufficient evidence to conclude that \(\mu>17\)
\(p-\text{value}=0.111\)