Math  /  Data & Statistics

QuestionA study was commissioned to find the mean weight of the residents in certain town. The study examined a random sample of 86 residents and found the mean weight to be 182 pounds with a standard deviation of 28 pounds. Use the normal distribution/empirical rule to estimate a 95%95 \% confidence interval for the mean, rounding all values to the nearest tenth.

Studdy Solution

STEP 1

1. The sample size is n=86 n = 86 .
2. The sample mean weight is xˉ=182 \bar{x} = 182 pounds.
3. The sample standard deviation is s=28 s = 28 pounds.
4. The distribution of weights is approximately normal.
5. We are using the empirical rule (68-95-99.7 rule) to estimate the confidence interval.

STEP 2

1. Determine the standard error of the mean.
2. Calculate the margin of error for a 95% confidence interval.
3. Compute the confidence interval.

STEP 3

Calculate the standard error of the mean (SEM) using the formula:
SEM=sn\text{SEM} = \frac{s}{\sqrt{n}}
Substitute the given values:
SEM=2886\text{SEM} = \frac{28}{\sqrt{86}}
Calculate the result:
SEM289.27363.0\text{SEM} \approx \frac{28}{9.2736} \approx 3.0

STEP 4

For a 95% confidence interval, the empirical rule suggests using approximately 2 standard deviations from the mean. Thus, calculate the margin of error (ME):
ME=2×SEM=2×3.0=6.0\text{ME} = 2 \times \text{SEM} = 2 \times 3.0 = 6.0

STEP 5

Calculate the confidence interval by adding and subtracting the margin of error from the sample mean:
Lower bound=xˉME=1826.0=176.0\text{Lower bound} = \bar{x} - \text{ME} = 182 - 6.0 = 176.0
Upper bound=xˉ+ME=182+6.0=188.0\text{Upper bound} = \bar{x} + \text{ME} = 182 + 6.0 = 188.0
Thus, the 95% confidence interval is:
(176.0,188.0)(176.0, 188.0)
The estimated 95% confidence interval for the mean weight is:
(176.0,188.0)\boxed{(176.0, 188.0)}

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord