An SRS of size n was taken to estimate mean body mass index (BMI) for girls between 13 and 19 years of age. The 95% confidence interval obtained had lower limit 19.5 and upper limit 26.3. Which of the following is NOT true? 1. A total of 95% of all teenage girls have BMI between 19.5 and 26.3. 2. The margin of error is 34. 3. The value z in the margin of error is 1.96. 4. A total of 95% of all SRS of size n contain the true mean BMI

Question

Mathematics

Lainey Wheeler

12 May, 20:01

An SRS of size n was taken to estimate mean body mass index (BMI) for girls between 13 and 19 years of age. The 95% confidence interval obtained had lower limit 19.5 and upper limit 26.3. Which of the following is NOT true? 1. A total of 95% of all teenage girls have BMI between 19.5 and 26.3. 2. The margin of error is 34. 3. The value z in the margin of error is 1.96. 4. A total of 95% of all SRS of size n contain the true mean BMI

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Memphis · Answer 1

Option 1 is the only untrue statement of the 4 statements.

The only wrong statement about the confidence interval above is the one about the total of 95% of all teenage girls having BMI between 19.5 and 26.3.

Step-by-step explanation:

The question provides that for the BMI of girls with age between 13 and 19, the 95% confidence interval has a lower limit of 19.5 and an upper limit of 26.3.

We will take the statement one after the other.

Statement 1: A total of 95% of all teenage girls have BMI between 19.5 and 26.3.

This is a wrong statement. It doesn't not follow the definition for confidence interval for a set of sample.

Rather confidence interval, expresses that the true value (mean) exists in the (lower limit, upper limit) range with a confidence level of 95%.

Statement 2: The margin of error is 34.

The margin of error is usually used to calculate the lower and upper limit of the confidence interval.

Basically, the interval is usually between

(Sample mean ± margin of error)

If the sample mean = xbar

And the margin of error = α

xbar - α = lower limit of the confidence interval = 19.5

xbar + α = upper limit of the confidence interval = 26.3

xbar - α = 19.5

xbar + α = 26.3

summing these together

xbar = (19.5+26.3) / 2 = 22.9

and the margin of error = (22.9 - 19.5) or (26.3 - 22.9) = 3.4.

So, this statement is correct!

Statement 3: The value z in the margin of error is 1.96.

The margin of error is given as the product of the z-multiplier (from the z-tables) and the sample standard deviation.

The z-multiplier for a 95% confidence interval, as obtained from literature and the z-tables is truly 1.96.

This statement is very true.

Statement 4: A total of 95% of all SRS of size n contain the true mean BMI.

Just like I described the meaning of confidence interval in the explanation under the first statement, this is as close to the meaning of confidence interval as can be. This statement is also very true.

Hence, only statement 1 is not correct of the 4 statements.