Answer :
To solve this problem, we need to obtain a 95% confidence interval for the difference in mean DEHP concentration between people who have eaten fast food in the last 24 hours and those who haven't. Here's how you can do it step-by-step:
1. Gather the Data:
- For participants who ate fast food:
- Number of participants, [tex]\( n_F = 3095 \)[/tex]
- Mean concentration, [tex]\( \bar{x}_F = 83.6 \, \text{ng/mL} \)[/tex]
- Standard deviation, [tex]\( s_F = 194.7 \, \text{ng/mL} \)[/tex]
- For participants who did not eat fast food:
- Number of participants, [tex]\( n_N = 5782 \)[/tex]
- Mean concentration, [tex]\( \bar{x}_N = 59.1 \, \text{ng/mL} \)[/tex]
- Standard deviation, [tex]\( s_N = 152.1 \, \text{ng/mL} \)[/tex]
2. Calculate the Mean Difference:
- The difference in sample means is:
[tex]\[
\text{Mean difference} = \bar{x}_F - \bar{x}_N = 83.6 - 59.1 = 24.5 \, \text{ng/mL}
\][/tex]
3. Calculate the Standard Error of the Difference:
- The standard error (SE) of the difference in means is calculated using the formula:
[tex]\[
SE_{\text{diff}} = \sqrt{\left(\frac{s_F^2}{n_F}\right) + \left(\frac{s_N^2}{n_N}\right)} = \sqrt{\left(\frac{194.7^2}{3095}\right) + \left(\frac{152.1^2}{5782}\right)}
\][/tex]
4. Determine the Z-score for a 95% Confidence Interval:
- For a 95% confidence level, a Z-score of approximately 1.96 is typically used.
5. Calculate the Confidence Interval:
- The confidence interval is calculated using the formula:
[tex]\[
\text{CI lower} = \text{Mean difference} - (z_{\text{score}} \times SE_{\text{diff}})
\][/tex]
[tex]\[
\text{CI upper} = \text{Mean difference} + (z_{\text{score}} \times SE_{\text{diff}})
\][/tex]
6. Result:
- The 95% confidence interval for the difference in mean concentration is [tex]\( (16.6, 32.4) \, \text{ng/mL} \)[/tex].
This interval provides us with a range in which we are 95% confident that the true difference in mean DEHP concentration between the two groups lies.
1. Gather the Data:
- For participants who ate fast food:
- Number of participants, [tex]\( n_F = 3095 \)[/tex]
- Mean concentration, [tex]\( \bar{x}_F = 83.6 \, \text{ng/mL} \)[/tex]
- Standard deviation, [tex]\( s_F = 194.7 \, \text{ng/mL} \)[/tex]
- For participants who did not eat fast food:
- Number of participants, [tex]\( n_N = 5782 \)[/tex]
- Mean concentration, [tex]\( \bar{x}_N = 59.1 \, \text{ng/mL} \)[/tex]
- Standard deviation, [tex]\( s_N = 152.1 \, \text{ng/mL} \)[/tex]
2. Calculate the Mean Difference:
- The difference in sample means is:
[tex]\[
\text{Mean difference} = \bar{x}_F - \bar{x}_N = 83.6 - 59.1 = 24.5 \, \text{ng/mL}
\][/tex]
3. Calculate the Standard Error of the Difference:
- The standard error (SE) of the difference in means is calculated using the formula:
[tex]\[
SE_{\text{diff}} = \sqrt{\left(\frac{s_F^2}{n_F}\right) + \left(\frac{s_N^2}{n_N}\right)} = \sqrt{\left(\frac{194.7^2}{3095}\right) + \left(\frac{152.1^2}{5782}\right)}
\][/tex]
4. Determine the Z-score for a 95% Confidence Interval:
- For a 95% confidence level, a Z-score of approximately 1.96 is typically used.
5. Calculate the Confidence Interval:
- The confidence interval is calculated using the formula:
[tex]\[
\text{CI lower} = \text{Mean difference} - (z_{\text{score}} \times SE_{\text{diff}})
\][/tex]
[tex]\[
\text{CI upper} = \text{Mean difference} + (z_{\text{score}} \times SE_{\text{diff}})
\][/tex]
6. Result:
- The 95% confidence interval for the difference in mean concentration is [tex]\( (16.6, 32.4) \, \text{ng/mL} \)[/tex].
This interval provides us with a range in which we are 95% confident that the true difference in mean DEHP concentration between the two groups lies.
