Question

Scientists estimate that there are 10 times more bacterial cells in your body than your own body's cells, and new studies on bacteria in the gut indicate that your gut microbes might be influencing you more than you realize, having positive or negative effects on health, development, and possibly even personality and behavior. A recent study \(^{26}\) found that the average number of unique genes in gut bacteria, for a sample of 99 healthy European individuals, was 564 million, with a standard deviation of 122 million. Use the t-distribution to find and interpret a \(95 \%\) confidence interval for the mean number of unique genes in gut bacteria for European individuals.

Short answer

The exact numerical calculations will provide a 95% confidence interval for the mean number of unique genes in gut bacteria for European individuals. This result means that we can be 95% confident that the interval contains the true mean number of unique genes in gut bacteria for all European individuals.

Step-by-step solution

Identify given values

The problem provides the following information: sample mean (\(x̄\)) = 564 million, sample standard deviation (s) = 122 million, sample size (n) = 99, and confidence level = 0.95 or 95%.

Determine the degrees of freedom

For a t-distribution, the degrees of freedom is typically calculated as the sample size minus 1. In this case, the degrees of freedom would be \(n - 1 = 99 - 1 = 98\).

Look up the t-value

Use a t-table or a calculator to find the t-value that corresponds to the 95% confidence level and 98 degrees of freedom. This t-value will be around 1.984, but its exact value might slightly vary depending on the table or calculator used.

Calculate the margin of error

Using the t-value, calculate the margin of error, which is the amount added and subtracted from the sample mean to create the confidence interval. The formula for the margin of error is \(E = t * s / \sqrt{n}\), where \(t\) is the t-score, \(s\) is the standard deviation, and \(n\) is the sample size. So, in this case, \(E \approx 1.984 * 122/\sqrt{99}\).

Calculate the confidence interval

The confidence interval is then calculated by subtracting and adding the margin of error from the sample mean. The formula used is \(x̄ \pm E\). Compute the exact numerical values and express the result in terms of millions.