Question

As part of a study of the development of the thymus gland, researchers weighed the glands of five chick embryos after 14 days of incubation. The thymus weights (mg) were as follows:

41.5, 39.2, 41.2, 42.3, 41.3.

Assume that thymus gland weight of chick embryos follow normal distribution. If you were to construct a 95 percent confidence interval for the mean thymus gland weight, it would take the form:

(sample mean) +/- t.quantile * (standard error of sample mean)

What is the correct value of t.quantile for a 95 percent confidence interval in this study?

A. 2.1319

B. 2.7764

C. 2.5706

D. 1.96

E. 2.0150

Answer 1

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(0.025,4)".And we see that
`t_(\alpha/2)=2.7764`

So the correct option would be:

B. 2.7764

Explanation:

Previous concepts

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".

`\bar X` represent the sample mean for the sample

`\mu` population mean (variable of interest)

s represent the sample standard deviation

n represent the sample size

Solution to the problem

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

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

In order to calculate the critical value
`t_(\alpha/2)` we need to find first the degrees of freedom, given by:

`df=n-1=5-1=4`

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(0.025,4)".And we see that
`t_(\alpha/2)=2.7764`

So the correct option would be:

B. 2.7764