If a researcher wanted to know the mean weight (the mean is the sum of all the measurements divided by the number of measurements) of women in the U.S., the weight of every woman would have to be measured and then the mean weight calculated, an impossible task. Instead, researchers find a representative sample of women and find the mean weight of the sample. Because the entire population of women is not used, there is a possibility that the calculated mean weight is not the true mean weight. For one study, suppose researchers used the formula
168 − ì
1.68
< 2.33,where ì is the true mean weight, in pounds, of all women, to be 98% sure of the range of values for the true mean weight. Using this inequality, what would be the range of mean weights of women in the U.S.? Round to the nearest tenth of a pound.
_____ lb < ì < _____ lb
Well, finding the mean weight of all women in the U.S. is certainly a hefty task. Let's calculate the range of mean weights using the given inequality.
We have the formula 168 - ì/√(1.68) < 2.33.
To isolate ì, we need to flip the inequality and perform some algebraic shenanigans.
First, we'll subtract 2.33 from both sides:
-2.33 < 168 - ì/√(1.68).
Next, we'll subtract 168 from both sides:
-170.33 < -ì/√(1.68).
To get rid of the negative on the left side, we'll multiply both sides by -1:
170.33 > ì/√(1.68).
Finally, we'll multiply both sides by √(1.68):
170.33√(1.68) > ì.
Now, let's calculate the range using our result:
To find the lower bound, we'll substitute ì with 170.33√(1.68):
ì > 170.33√(1.68).
And for the upper bound, we'll use the same result:
ì < 170.33√(1.68).
Now we can round our answers to the nearest tenth of a pound:
The range of mean weights for women in the U.S. is approximately 170.3 lb < ì < 263.6 lb.
Keep in mind, though, that these calculations are based on the given formula and assumptions. So take them with a pinch of humor!
To find the range of mean weights of women in the U.S., we will use the given formula:
168 - (1.68 / √n) < ì < 168 + (1.68 / √n)
In this formula, ì represents the true mean weight of all women in pounds, and n represents the sample size.
To be 98% sure of the range of values for the true mean weight, we need to find the value of z that corresponds to a confidence level of 98%. This can be done using a standard normal distribution table or calculator. The value of z for 98% confidence level is approximately 2.33.
Now, we can substitute this value of z into the formula:
168 - (1.68 / √n) < ì < 168 + (1.68 / √n)
Rounding to the nearest tenth of a pound, the range of mean weights of women in the U.S. is:
168 - (1.68 / √n) lb < ì < 168 + (1.68 / √n) lb
To find the range of mean weights of women in the U.S., we can use the formula:
168 - (1.68 / √n) < ì < 168 + (1.68 / √n)
In this formula, ì represents the true mean weight, and n represents the sample size.
In this case, the formula given is 168 - (1.68 / √n) < ì < 168 + (1.68 / √n), with a confidence level of 98%.
To determine the range, we need to find the critical value associated with a confidence level of 98%. The critical value is the number of standard deviations away from the mean needed to achieve the desired confidence level. In this case, the critical value is 2.33.
From the formula, we can set up the following inequality:
168 - (1.68 / √n) < ì < 168 + (1.68 / √n)
Using the given critical value of 2.33, we can set up the following inequality:
168 - (1.68 / √n) < ì < 168 + (1.68 / √n) < 168 + (2.33 / √n)
Next, we solve for the lower bound:
168 - (1.68 / √n) < ì
Subtract 168 from both sides to isolate ì:
- (1.68 / √n) < ì - 168
Multiply both sides by -1 to flip the inequality:
(1.68 / √n) > 168 - ì
Finally, solving for the upper bound, we have:
ì < 168 + (1.68 / √n) < ì < 168 + (2.33 / √n)
Therefore, the range of mean weights of women in the U.S. is:
168 - (1.68 / √n) < ì < ì < 168 + (2.33 / √n)
Please note that to determine the actual range, we would need the sample size, as it is necessary for the calculation.