SOLUTION: The weight in pounds of a random sample of 7 babies were recorded before and after taking a certain vitamin syrup for a period of one month. a) Construct a 95% confidence interv

Question 1192384: The weight in pounds of a random sample of 7 babies were recorded before and after
taking a certain vitamin syrup for a period of one month.
a) Construct a 95% confidence interval of the difference of the mean weights.
b) Is there a significant difference in their weights at 0.05 level?
before 29 22 25 29 26 24 31
after 39 26 25 35 33 36 32
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

Part a)

Hypothesis:
H0:

H1:

This is a two-tailed test.

Given Data

before	after
29	39
22	26
25	25
29	35
26	33
24	36
31	32

For each row separately, subtract the values in the order A-B to form a column of differences (d)

A	B	A-B
before	after	d = difference
29	39	-10
22	26	-4
25	25	0
29	35	-6
26	33	-7
24	36	-12
31	32	-1

Using a calculator, compute the sample mean and sample standard deviation of the list of difference values in that new column.
You should get

for the sample mean and

for the sample standard deviation.

The sample size is n = 7 which means the degrees of freedom (df) are
df = n-1
df = 7-1
df = 6
Refer to a T table like this one
https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf
Locate the row that has df = 6
Locate the column that has 0.05 as the two tails
Intersecting the row and column yields the approximate value t = 2.447

Now let's compute the margin of error.
E = margin of error

We can then find the lower bound L

And the upper bound (U) as well

The 95% confidence interval in the format (L, U) is roughly (-9.81, -1.62)
Negative values are valid because it represents the idea that the "after" weights are larger than the "before" ones.

=====================================================================================

Part b)

The result of part a) is equivalent to saying we are 95% confident that

when using the format

In other words, we are 95% confident that the pameter we're after (

) is somewhere between -9.81 and -1.62

Is

in this confidence interval? No, it is not.
So it's unlikely that

is the case.
There appears to be a significant difference in the weights.

This means we'd reject the null and side with the alternative hypothesis to conclude that

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