Question

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

9.7 9.9 10.3 10.1 10.5 9.4 9.9 10.1 9.7 10.3

a. Find the sample mean and standard deviation for these data.
b. Assume that the historical average is true. Calculate the observed value of the t-statistic.
c. What is the probability of these statistics (or worse) if the true mean were 10.1 mm?

Answer 1

a) Sample mean = 9.99

Sample standard deviation = 0.3348

b) -1.0389

c) 0.1631

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(99.9)/(10) = 9.99`

Sum of squares of differences = 1.009

`S.D = \sqrt{(1.009)/(9)} = 0.3348`

b) observed value of the t-statistic

Formula:

`t_(stat) = \displaystyle\frac{\bar{x} - \mu}{(\sigma)/(√(n)) }`

Putting all the values, we have

`t_(stat) = \displaystyle(9.99 - 10.1)/((0.3348)/(√(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