Question

the work week for adults in the us that work full time is normally distributed with a mean of 47 hours. a newly hired engineer at a start - up company believes that employees at start - up companies work more on average then most working adults in the us. she asks 12 engineering friends at start - ups for the lengths in hours of their work week. their responses are shown in the table below. test the claim using a 1% level of significance. give answer to at least 4 decimal places.
hours
48
47
59
50
49
63
47
45
47
46
52
51

Explanation:

Step1: Calculate sample mean $\bar{x}$

$\bar{x}=\frac{48 + 47+59+50+49+63+47+45+47+46+52+51}{12}=\frac{594}{12}=49.5$

Step2: Calculate sample standard - deviation $s$

First, calculate the squared differences from the mean for each data - point:
$(48 - 49.5)^2=(-1.5)^2 = 2.25$, $(47 - 49.5)^2=(-2.5)^2 = 6.25$, $(59 - 49.5)^2=(9.5)^2 = 90.25$, $(50 - 49.5)^2=(0.5)^2 = 0.25$, $(49 - 49.5)^2=(-0.5)^2 = 0.25$, $(63 - 49.5)^2=(13.5)^2 = 182.25$, $(47 - 49.5)^2=(-2.5)^2 = 6.25$, $(45 - 49.5)^2=(-4.5)^2 = 20.25$, $(47 - 49.5)^2=(-2.5)^2 = 6.25$, $(46 - 49.5)^2=(-3.5)^2 = 12.25$, $(52 - 49.5)^2=(2.5)^2 = 6.25$, $(51 - 49.5)^2=(1.5)^2 = 2.25$
The sum of squared differences $\sum_{i = 1}^{n}(x_i-\bar{x})^2=2.25 + 6.25+90.25+0.25+0.25+182.25+6.25+20.25+6.25+12.25+6.25+2.25 = 335$
$s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}=\sqrt{\frac{335}{11}}\approx5.5055$

Step3: State the hypotheses

$H_0:\mu\leq47$ (null hypothesis), $H_1:\mu>47$ (alternative hypothesis)

Step4: Calculate the test - statistic

The test - statistic for a one - sample t - test is $t=\frac{\bar{x}-\mu}{s/\sqrt{n}}$, where $\mu = 47$, $\bar{x}=49.5$, $s\approx5.5055$, and $n = 12$
$t=\frac{49.5 - 47}{5.5055/\sqrt{12}}\approx1.5675$

Step5: Determine the critical value

The degrees of freedom is $df=n - 1=12 - 1 = 11$. For a one - tailed test with $\alpha = 0.01$, the critical value $t_{\alpha,df}=t_{0.01,11}= 2.7181$

Step6: Make a decision

Since $t = 1.5675

Answer:

We fail to reject the null hypothesis. The test - statistic is approximately $t = 1.5675$ and the critical value is $t_{0.01,11}=2.7181$.