Question

Researchers are interested in the effect of a certain nutrient on the growth rate of plant seedlings. Using a hydroponics grow procedure that utilized water containing the nutrient, they planted six tomato plants and recorded the heights of each plant 14 days after germination. Those heights, measured in millimeters, were as follows:

53.1, 60.2, 60.6, 62.1, 64.4, 68.6.
1. Given a margin of error of 5.4 mm, construct a 95% confidence interval for the population mean height.
2. What is the point estimate of the population mean height of this variety of seedling 14 days after germination? Do not round your final answer.

Answer 1

1)
`\bar X \pm ME`

And the margin of error for this case is
`ME= 5.4`

And the confidence interval would be:

`61.5-5.4 = 56.1`

`61.5+5.4 = 66.9`

2) For this case the best estimator for the population mean
`\mu` is given by the sample mean:

`\bar X = 61.5`

And the reason of this is beacuse is an unbiased estimator of the parameter:

`E(\bar X) = \mu`

Explanation:

Data given:

53.1, 60.2, 60.6, 62.1, 64.4, 68.6.

We can calculate the sample mean with the following formula:

`\bar X = (\sum_(i=1)^n X_i)/(n)`

`\bar X= 61.5` represent the sample mean for the sample

`\mu` population mean (variable of interest)

s represent the sample standard deviation

n represent the sample size

Part 1

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

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

Or equivalently:

`\bar X \pm ME`

And the margin of error for this case is
`ME= 5.4`

And the confidence interval would be:

`61.5-5.4 = 56.1`

`61.5+5.4 = 66.9`

Part 2

For this case the best estimator for the population mean
`\mu` is given by the sample mean:

`\bar X = 61.5`

And the reason of this is beacuse is an unbiased estimator of the parameter:

`E(\bar X) = \mu`