Here are summary statistics for randomly selected weights of newborn girls: n=225, x=33.6hg, s=7.4 hg. Construct a confidence interval estimate of the mean. Use a 99% confidence level. Are these results very different from the confidence interval 31.6 hg< u < 36.0hg with only 17 sample values, x=33.8hg, and s=3.1hg?
What is the confidence interval for the population mean u?
Are the results between the two confidence intervals very different?
To construct a confidence interval estimate of the mean weight of newborn girls, we can use the formula:
Confidence Interval = x ± (Z * (s/√n))
where:
n = sample size
x = sample mean
s = sample standard deviation
Z = Z-score corresponding to the desired confidence level
Given the following values:
n = 225
x = 33.6 hg
s = 7.4 hg
Confidence level = 99%
To find the Z-score for a 99% confidence level, we can use a standard normal distribution table or a statistical calculator. The Z-score for a 99% confidence level is approximately 2.575.
Plugging the values into the formula:
Confidence Interval = 33.6 ± (2.575 * (7.4/√225))
Calculating the confidence interval:
Confidence Interval = 33.6 ± (2.575 * 0.492)
Confidence Interval = 33.6 ± 1.267
Confidence Interval = (32.333, 34.867)
The confidence interval estimate for the mean weight of newborn girls is (32.333 hg, 34.867 hg) at a 99% confidence level.
Now, let's compare this result with the given confidence interval for the population mean with the smaller sample size values:
Confidence Interval = 31.6 hg < u < 36.0 hg
Sample size (n) = 17
Sample mean (x) = 33.8 hg
Sample standard deviation (s) = 3.1 hg
Comparing the two confidence intervals, we can see that the range of the first confidence interval (32.333 hg to 34.867 hg) falls within the range of the second confidence interval (31.6 hg to 36.0 hg). Therefore, the results are not very different.
To construct a confidence interval estimate of the mean, we will use the following information from the given data:
- Sample size (n) = 225
- Sample mean (x̄) = 33.6 hg
- Sample standard deviation (s) = 7.4 hg
- Confidence level = 99%
To calculate the confidence interval, we can use the formula:
Confidence Interval = x̄ ± (Critical Value * (s/√n))
1. Calculate the critical value:
Since the confidence level is 99%, we need to find the critical value that corresponds to a 99% confidence level. This critical value can be obtained from a standard normal distribution table or by using a calculator. For a 99% confidence level, the critical value is approximately 2.576.
2. Calculate the margin of error:
The margin of error is obtained by multiplying the critical value by the standard error, which is the standard deviation divided by the square root of the sample size: (s/√n)
Margin of Error = 2.576 * (7.4/√225)
3. Calculate the confidence interval:
Using the formula mentioned earlier, we can calculate the confidence interval estimate:
Confidence Interval = 33.6 ± (2.576 * (7.4/√225))
After performing the calculations, you will find the confidence interval for the population mean (u) to be:
Confidence Interval = 33.6 ± 1.007 (rounded to one decimal place)
Confidence Interval = 32.6 to 34.6 hg
Regarding the comparison between the two confidence intervals:
The first confidence interval is from a sample size of 225, while the second confidence interval is from a sample size of 17. Generally, a larger sample size leads to a more precise estimate.
The second confidence interval has a smaller range (31.6 to 36.0 hg) compared to the first confidence interval (32.6 to 34.6 hg). This suggests that the second confidence interval is narrower, indicating more precision or less variability in the data. However, we cannot definitively conclude that the results are "very different" based solely on the confidence intervals. It would be more appropriate to conduct hypothesis testing to determine if there is a statistically significant difference between the means.