Question

The true average diameter of ball bearings of a certain type is supposed to be 0.05 in. A one-sample t-test will be carried out to see whether this is the case.

What conclusion is appropriate in each of the following situations?
a. n=13, t=1.6, α=0.05.
b. n=13, t=−1.6, α=0.05.
c. n=25, t=−2.6, α=0.01.

Answer 1

a) H0:
`\mu \leq \mu_o`

H1:
`\mu > \mu_o`

n = 13 represent the sample size

`t = 1.6` represent the calculated statistic

The degrees of freedom are given by:

`df = n-1 = 13-1=12`

We can calculathe the p value with this formula:

`p_v = P(t_(12) >1.6) = 0.068`

Since
`p_v >\alpha` we fail to reject the null hypothesis on this case at 5% of significance.

b) H0:
`\mu \geq \mu_o`

H1:
`\mu < \mu_o`

n = 13 represent the sample size

`t = -1.6` represent the calculated statistic

The degrees of freedom are given by:

`df = n-1 = 13-1=12`

We can calculathe the p value with this formula:

`p_v = P(t_(12) <-1.6) = 0.068`

Since
`p_v >\alpha` we fail to reject the null hypothesis on this case at 5% of significance.

c) H0:
`\mu \geq \mu_o`

H1:
`\mu < \mu_o`

n = 25 represent the sample size

`t = -2.6` represent the calculated statistic

The degrees of freedom are given by:

`df = n-1 = 25-1=24`

We can calculathe the p value with this formula:

`p_v = P(t_(24) <-2.6) = 0.0078`

Since
`p_v <\alpha` we can reject the null hypothesis on this case at 1% of significance.

Explanation:

Part a

For this case we assume that we are testing the following system of hypothesis

H0:
`\mu \leq \mu_o`

H1:
`\mu > \mu_o`

n = 13 represent the sample size

`t = 1.6` represent the calculated statistic

The degrees of freedom are given by:

`df = n-1 = 13-1=12`

We can calculathe the p value with this formula:

`p_v = P(t_(12) >1.6) = 0.068`

Since
`p_v >\alpha` we fail to reject the null hypothesis on this case at 5% of significance.

Part b

For this case we assume that we are testing the following system of hypothesis

H0:
`\mu \geq \mu_o`

H1:
`\mu < \mu_o`

n = 13 represent the sample size

`t = -1.6` represent the calculated statistic

The degrees of freedom are given by:

`df = n-1 = 13-1=12`

We can calculathe the p value with this formula:

`p_v = P(t_(12) <-1.6) = 0.068`

Since
`p_v >\alpha` we fail to reject the null hypothesis on this case at 5% of significance.

Part c

For this case we assume that we are testing the following system of hypothesis

H0:
`\mu \geq \mu_o`

H1:
`\mu < \mu_o`

n = 25 represent the sample size

`t = -2.6` represent the calculated statistic

The degrees of freedom are given by:

`df = n-1 = 25-1=24`

We can calculathe the p value with this formula:

`p_v = P(t_(24) <-2.6) = 0.0078`

Since
`p_v <\alpha` we can reject the null hypothesis on this case at 1% of significance.