The number of grams of carbohydrates in a 12-ounce serving of a regular soft drink is listed here for a random sample of sodas. Find the 99% confidence interval for the mean number of carbohydrates in all brands of soda.

48 37 52 40 43 46 41 38
41 45 45 33 35 52 45 41

To find the 99% confidence interval for the mean number of carbohydrates in all brands of soda, follow these steps:

Step 1: Calculate the sample mean.
The sample mean (x̄) can be calculated by summing up all the values in the sample and dividing by the number of data points.
x̄ = (48 + 37 + 52 + 40 + 43 + 46 + 41 + 38 + 41 + 45 + 45 + 33 + 35 + 52 + 45 + 41) / 16
x̄ = 46.125

Step 2: Calculate the sample standard deviation.
The sample standard deviation (s) can be calculated using the formula:
s = sqrt(Σ(xi - x̄)^2 / (n-1))
where xi is each value in the sample, x̄ is the sample mean, and n is the number of data points.
For our sample data:

Σ(xi - x̄)^2 = (48-46.125)^2 + (37-46.125)^2 + (52-46.125)^2 + (40-46.125)^2 + (43-46.125)^2 + (46-46.125)^2 + (41-46.125)^2 + (38-46.125)^2 + (41-46.125)^2 + (45-46.125)^2 + (45-46.125)^2 + (33-46.125)^2 + (35-46.125)^2 + (52-46.125)^2 + (45-46.125)^2 + (41-46.125)^2
Σ(xi - x̄)^2 = 828.375

Using the formula, we can calculate the sample standard deviation:
s = sqrt(Σ(xi - x̄)^2 / (n-1))
s = sqrt(828.375 / (16-1))
s = sqrt(828.375 / 15)
s = sqrt(55.225)
s ≈ 7.43

Step 3: Calculate the margin of error.
The margin of error (E) can be calculated using the formula:
E = t * (s / sqrt(n))
where t is the critical value for a desired confidence level, s is the sample standard deviation, and n is the number of data points in the sample.
For a 99% confidence level with 16 data points, the critical value can be found using a t-table or calculator. In this case, the critical value is approximately 2.921.
E = 2.921 * (7.43 / sqrt(16))
E = 2.921 * (7.43 / 4)
E ≈ 5.36

Step 4: Calculate the confidence interval.
The confidence interval can be calculated using the formula:
CI = x̄ ± E
CI = 46.125 ± 5.36
CI ≈ (40.765, 51.485)

Therefore, at a 99% confidence level, the mean number of carbohydrates in all brands of soda is estimated to be between approximately 40.765 and 51.485 grams.

To find the 99% confidence interval for the mean number of carbohydrates in all brands of soda, we can use the formula:

Confidence Interval = Sample Mean ± (Critical Value) x (Standard Error)

1. Calculate the sample mean:
Add up all the values and divide by the number of values in the sample.
Sample Mean = (48 + 37 + 52 + 40 + 43 + 46 + 41 + 38 + 41 + 45 + 45 + 33 + 35 + 52 + 45 + 41) / 16

2. Calculate the standard deviation:
Find the standard deviation of the sample by calculating the square root of the variance.
Standard Deviation = sqrt(((48 - Sample Mean)^2 + (37 - Sample Mean)^2 + ... + (41 - Sample Mean)^2) / (n - 1))

3. Calculate the standard error:
Divide the standard deviation by the square root of the sample size.
Standard Error = Standard Deviation / sqrt(n)

4. Determine the critical value:
Since we want a 99% confidence interval, we need to find the critical value associated with a 99% confidence level.
You can obtain this value from a standard normal distribution table or use a statistical calculator/software.
The critical value for a 99% confidence level is approximately 2.617.

5. Calculate the confidence interval:
Confidence Interval = Sample Mean ± (Critical Value) x (Standard Error)

Now, let's plug in the values:
Sample Mean = (48 + 37 + 52 + 40 + 43 + 46 + 41 + 38 + 41 + 45 + 45 + 33 + 35 + 52 + 45 + 41) / 16
Standard Deviation = sqrt(((48 - Sample Mean)^2 + (37 - Sample Mean)^2 + ... + (41 - Sample Mean)^2) / (n - 1))
Standard Error = Standard Deviation / sqrt(n)
Critical Value = 2.617

Confidence Interval = Sample Mean ± (Critical Value) x (Standard Error)

Finally, calculate the confidence interval using the values derived above.

Hope this helps:

Mean 42.625
Standard Error 1.37196635
Median 42
Mode 41
Standard Deviation 5.487865402
Sample Variance 30.11666667
Kurtosis -0.413446348
Skewness 0.110205125
Range 19
Minimum 33
Maximum 52
Sum 682
Count 16
Confidence Level(99.0%) 4.04279092

This is really easy to do. Just type the data into excel and then go to "Data-Data Analysis (make sure you have enabled this)" Then go to "Descriptive Statistics and enter in the C.I. and anything else. Good Luck