Question

A chemist made eight independent measurements of the melting point of tungsten. She obtained a sample mean of 3410.14 degrees Celsius and a sample standard deviation of 1.018 degrees. a) Use the Student’s t distribution to find a 95% confidence interval for the melting point of tungsten. b) Use the Student’s t distribution to find a 98% confidence interval for the melting point of tungsten.

Answer 1

a) The 95% confidence interval would be given by (3409.291;3410.989)

b) The 98% confidence interval would be given by (3409.060;3411.220)

Explanation:

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

Part a

`\bar X=3410.14` represent the sample mean

`\mu` population mean (variable of interest)

`s=1.018` represent the sample standard deviation

n=8 represent the sample size

95% confidence interval

The confidence interval for the mean is given by the following formula:

`\bar X \pm t_(\alpha/2)(s)/(√(n))` (1)

First we need to find the degrees of freedom given by:

`df=n-1=8-1=7`

Since the Confidence is 0.95 or 95%, the value of
`\alpha=0.05` and
`\alpha/2 =0.025`, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,7)".And we see that
`t_(\alpha/2)=2.36`

Now we have everything in order to replace into formula (1):

`3410.14-2.36(1.018)/(√(8))=3409.291`

`3410.14+2.36(1.018)/(√(8))=3410.989`

So on this case the 95% confidence interval would be given by (3409.291;3410.989)

Part b

Since the Confidence is 0.98 or 98%, the value of
`\alpha=0.02` and
`\alpha/2 =0.01`, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.01,7)".And we see that
`t_(\alpha/2)=3.00`

Now we have everything in order to replace into formula (1):

`3410.14-3.00(1.018)/(√(8))=3409.060`

`3410.14+3.00(1.018)/(√(8))=3411.220`

So on this case the 98% confidence interval would be given by (3409.060;3411.220)