Question
page 15 of 18
Interpreting the Margin of Error
When a random sample, n
, is taken from a large population, the margin of error can be approximated using the following formula.
Margin of Error: ±1n√
Use the results of the sample and the margin of error to write an interval in which the parameter lies within a 95%
confidence interval. This lesson will only use a 95%
confidence interval. The interval can be represented as
(p−1n√, p+1n√)
where p
represents the percent of the sample with a certain characteristic.
Before interpreting the margin of error, complete the interactive element to practice calculating the margin of error and confidence interval.
A survey is conducted for a sample size n and finds that p% of participants get at least 8 hours of sleep each night. Calculate the margin of error for each scenario. Give each answer as a percent rounded to the nearest tenth.
If 34 individuals are surveyed and 91% get at least 8 hours of sleep each night, then the margin of error is
. If 97 individuals are surveyed and 82% get at least 8 hours of sleep each night, then the margin of error is
. If 245 individuals are surveyed and 67% get at least 8 hours of sleep each night, then the margin of error is
. If 598 individuals are surveyed and 71% get at least 8 hours of sleep each night, then the margin of error is
.
Interpreting the Margin of Error
When a random sample, n
, is taken from a large population, the margin of error can be approximated using the following formula.
Margin of Error: ±1n√
Use the results of the sample and the margin of error to write an interval in which the parameter lies within a 95%
confidence interval. This lesson will only use a 95%
confidence interval. The interval can be represented as
(p−1n√, p+1n√)
where p
represents the percent of the sample with a certain characteristic.
Before interpreting the margin of error, complete the interactive element to practice calculating the margin of error and confidence interval.
A survey is conducted for a sample size n and finds that p% of participants get at least 8 hours of sleep each night. Calculate the margin of error for each scenario. Give each answer as a percent rounded to the nearest tenth.
If 34 individuals are surveyed and 91% get at least 8 hours of sleep each night, then the margin of error is
. If 97 individuals are surveyed and 82% get at least 8 hours of sleep each night, then the margin of error is
. If 245 individuals are surveyed and 67% get at least 8 hours of sleep each night, then the margin of error is
. If 598 individuals are surveyed and 71% get at least 8 hours of sleep each night, then the margin of error is
.
Answers
GPT-4o mini
To calculate the margin of error for each scenario using the formula:
\[
\text{Margin of Error} = \pm \frac{1}{\sqrt{n}}
\]
where \( n \) is the sample size. The margin of error can also be expressed as a percentage by multiplying the resulting value by 100.
Let's calculate the margin of error for each scenario:
1. **For 34 individuals surveyed (p = 91% = 0.91)**:
\[
\text{Margin of Error} = \pm \frac{1}{\sqrt{34}}
\]
\[
\sqrt{34} \approx 5.831
\]
\[
\text{Margin of Error} \approx \pm \frac{1}{5.831} \approx \pm 0.171
\]
In percent:
\[
\text{Margin of Error} \approx 0.171 \times 100 \approx 17.1\%
\]
2. **For 97 individuals surveyed (p = 82% = 0.82)**:
\[
\text{Margin of Error} = \pm \frac{1}{\sqrt{97}}
\]
\[
\sqrt{97} \approx 9.848
\]
\[
\text{Margin of Error} \approx \pm \frac{1}{9.848} \approx \pm 0.102
\]
In percent:
\[
\text{Margin of Error} \approx 0.102 \times 100 \approx 10.2\%
\]
3. **For 245 individuals surveyed (p = 67% = 0.67)**:
\[
\text{Margin of Error} = \pm \frac{1}{\sqrt{245}}
\]
\[
\sqrt{245} \approx 15.652
\]
\[
\text{Margin of Error} \approx \pm \frac{1}{15.652} \approx \pm 0.064
\]
In percent:
\[
\text{Margin of Error} \approx 0.064 \times 100 \approx 6.4\%
\]
4. **For 598 individuals surveyed (p = 71% = 0.71)**:
\[
\text{Margin of Error} = \pm \frac{1}{\sqrt{598}}
\]
\[
\sqrt{598} \approx 24.485
\]
\[
\text{Margin of Error} \approx \pm \frac{1}{24.485} \approx \pm 0.041
\]
In percent:
\[
\text{Margin of Error} \approx 0.041 \times 100 \approx 4.1\%
\]
**Final Results:**
1. For 34 individuals surveyed: Margin of Error ≈ **17.1%**
2. For 97 individuals surveyed: Margin of Error ≈ **10.2%**
3. For 245 individuals surveyed: Margin of Error ≈ **6.4%**
4. For 598 individuals surveyed: Margin of Error ≈ **4.1%**
\[
\text{Margin of Error} = \pm \frac{1}{\sqrt{n}}
\]
where \( n \) is the sample size. The margin of error can also be expressed as a percentage by multiplying the resulting value by 100.
Let's calculate the margin of error for each scenario:
1. **For 34 individuals surveyed (p = 91% = 0.91)**:
\[
\text{Margin of Error} = \pm \frac{1}{\sqrt{34}}
\]
\[
\sqrt{34} \approx 5.831
\]
\[
\text{Margin of Error} \approx \pm \frac{1}{5.831} \approx \pm 0.171
\]
In percent:
\[
\text{Margin of Error} \approx 0.171 \times 100 \approx 17.1\%
\]
2. **For 97 individuals surveyed (p = 82% = 0.82)**:
\[
\text{Margin of Error} = \pm \frac{1}{\sqrt{97}}
\]
\[
\sqrt{97} \approx 9.848
\]
\[
\text{Margin of Error} \approx \pm \frac{1}{9.848} \approx \pm 0.102
\]
In percent:
\[
\text{Margin of Error} \approx 0.102 \times 100 \approx 10.2\%
\]
3. **For 245 individuals surveyed (p = 67% = 0.67)**:
\[
\text{Margin of Error} = \pm \frac{1}{\sqrt{245}}
\]
\[
\sqrt{245} \approx 15.652
\]
\[
\text{Margin of Error} \approx \pm \frac{1}{15.652} \approx \pm 0.064
\]
In percent:
\[
\text{Margin of Error} \approx 0.064 \times 100 \approx 6.4\%
\]
4. **For 598 individuals surveyed (p = 71% = 0.71)**:
\[
\text{Margin of Error} = \pm \frac{1}{\sqrt{598}}
\]
\[
\sqrt{598} \approx 24.485
\]
\[
\text{Margin of Error} \approx \pm \frac{1}{24.485} \approx \pm 0.041
\]
In percent:
\[
\text{Margin of Error} \approx 0.041 \times 100 \approx 4.1\%
\]
**Final Results:**
1. For 34 individuals surveyed: Margin of Error ≈ **17.1%**
2. For 97 individuals surveyed: Margin of Error ≈ **10.2%**
3. For 245 individuals surveyed: Margin of Error ≈ **6.4%**
4. For 598 individuals surveyed: Margin of Error ≈ **4.1%**