Question

Historically, these bolts have an average thickness of 10.1 mm. A recent random sample of 10 bolts yielded these thicknesses:

Answer 1

Answer:

a) Sample mean = 9.99

Sample standard deviation = 0.3348

b) -1.0389

c) 0.1631

Step-by-step explanation:

We are given the following in the question:

9.7, 9.9, 10.3, 10.1, 10.5, 9.4, 9.9, 10.1, 9.7, 10.3

a) sample mean and standard deviation

Formula:

$\text{Standard Deviation} = \sqrt{\displaystyle\frac{\sum (x_i -\bar{x})^2}{n-1}}$  

where $x_i$ are data points, $\bar{x}$ is the mean and n is the number of observations.  

$Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}$

$Mean =\displaystyle\frac{99.9}{10} = 9.99$

Sum of squares of differences = 1.009

$S.D = \sqrt{\dfrac{1.009}{9}} = 0.3348$

b) observed value of the t-statistic

Formula:

$t_{stat} = \displaystyle\frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}} }$

Putting all the values, we have

$t_{stat} = \displaystyle\frac{9.99 - 10.1}{\frac{0.3348}{\sqrt{10}} } = -1.0389$

c) probability of these statistics (or worse) if the true mean were 10.1 mm

Degree of freedom =  n - 1 = 9

Calculating the value from the table

$P(x < 9.99) = 0.1631$

0.1631 is the the probability of these statistics (or worse) if the true mean were 10.1 mm