Question

A student researcher measured the cadmium (Cd2+) content in a standard reference material (SRM) using graphite furnace atomic absorption spectrometry (GFAAS). With 7 replicate measurements, the researcher determined a mean Cd2 concentration of 2.899 ppb with a standard deviation of 0.001 ppb. Calculate the 99% confidence interval for this sample. Enter your answer with three significant figures.

Answer 1

To calculate the 99% confidence interval for the cadmium content in the sample, we can follow these steps:

1. Identify the sample size (n), the sample mean (mean), the sample standard deviation (std_dev), and the confidence level.

From the given information:
- Sample size, n = 7
- Sample mean, mean = 2.899 ppb
- Sample standard deviation, std_dev = 0.001 ppb
- Confidence level = 99%

2. Calculate the degrees of freedom (df). Since the sample size is n, the degrees of freedom will be df = n - 1.

Degrees of freedom, df = 7 - 1 = 6

3. Find the t-value that corresponds to the 99% confidence level with df degrees of freedom. To do this, we look up the t-table or use a statistical tool (like a software or calculator). The 99% confidence level refers to the probability that the true mean lies within the calculated interval, so we need the t-value for a two-tailed test. This means we want the t-value that will give us an area of (1-0.99)/2 = 0.005 in the tails of the t-distribution.

For df = 6, the t-value for a 99% confidence interval, taking into account both tails of the distribution, can be found using such a statistical tool.

4. Calculate the margin of error (E) using the formula:
E = t_value * (std_dev / sqrt(n))

5. Calculate the lower and upper limits of the confidence interval using the formulae:
- Lower limit = mean - E
- Upper limit = mean + E

6. Round the results to three significant figures.

Let's assume we have found the t-value for a 99% confidence interval with df = 6. Let's call this t_value.

Now, we calculate the margin of error E:

E = t_value * (std_dev / sqrt(n))

Substitute the known values:

E = t_value * (0.001 / sqrt(7))
E = t_value * (0.001 / 2.64575)
E = t_value * 0.0003779645

Next, we would calculate the confidence interval limits:

Lower limit = mean - E
Upper limit = mean + E

Substitute the mean and calculated E:

Lower limit = 2.899 - (t_value * 0.0003779645)
Upper limit = 2.899 + (t_value * 0.0003779645)

The final step is to round the lower and upper limits to three significant figures.

Without the actual t-value, we cannot compute the final confidence interval. Once you have the t-value, you can plug it into the formula to get the exact lower and upper bounds. Then round the results to maintain three significant figures as required.

Answer 2

The 99% confidence interval for this sample mean is between 2.895 ppb and 2.903 ppb.

Explanation:

We have the standard deviation for the sample, so we use the t-distribution to solve this question.

The first step to solve this problem is finding how many degrees of freedom, we have. This is the sample size subtracted by 1. So

df = 7 - 1 = 6

99% confidence interval

Now, we have to find a value of T, which is found looking at the t table, with 6 degrees of freedom(y-axis) and a confidence level of
`1 - (1 - 0.99)/(2) = 0.995`. So we have T = 3.7074

The margin of error is:

M = T*s = 3.7074*0.001 = 0.004

In which s is the standard deviation of the sample.

The lower end of the interval is the sample mean subtracted by M. So it is 2.899 - 0.004 = 2.895 ppb.

The upper end of the interval is the sample mean added to M. So it is 2.899 + 0.004 = 2.903 ppb.

The 99% confidence interval for this sample mean is between 2.895 ppb and 2.903 ppb.