Questions: A researcher wants to know how long it takes, on average, for a certain species of bacteria to divide. She watches 34 cells through a microscope and times how long it takes them to divide. She obtains the following data, in hours: 8.2, 8.5, 8.2, 7.5, 6.8, 8.2, 8.3, 7.8, 6.2, 8.2, 7.3, 7.6, 7.1, 5.1, 6.4, 6, 6.8, 6.3, 8.6, 7.3, 6.9, 7.9, 6.4, 7.4, 8, 7.4, 7.8, 8.1, 8.8, 8.4, 7.8, 8.8, 7.5, 7.8 Construct a 92% confidence interval for the average time it takes this species of bacteria to divide. We are 92% confident that this species of bacteria takes, on average, between 7.249 hours and 7.773 hours to divide.

A researcher wants to know how long it takes, on average, for a certain species of bacteria to divide. She watches 34 cells through a microscope and times how long it takes them to divide. She obtains the following data, in hours:
8.2, 8.5, 8.2, 7.5, 6.8, 8.2, 8.3, 7.8, 6.2, 8.2, 7.3, 7.6, 7.1, 5.1, 6.4, 6, 6.8, 6.3, 8.6, 7.3, 6.9, 7.9, 6.4, 7.4, 8, 7.4, 7.8, 8.1, 8.8, 8.4, 7.8, 8.8, 7.5, 7.8

Construct a 92% confidence interval for the average time it takes this species of bacteria to divide.

We are 92% confident that this species of bacteria takes, on average, between 7.249 hours and 7.773 hours to divide.

Transcript text: A researcher wants to know how long it takes, on average, for a certain species of bacteria to divide. She watches 34 cells through a microscope and times how long it takes them to divide. She obtains the following data, in hours: $8.2,8.5,8.2,7.5,6.8,8.2,8.3,7.8,6.2,8.2,7.3,7.6,7.1,5.1,6.4,6,6.8,6.3,8.6,7.3,6.9,7.9,6.4, 7.4,8,7.4,7.8,8.1,8.8,8.4,7.8,8.8,7.5,7.8$ Construct a $92 \%$ confidence interval for the average time it takes this species of bacteria to divide. We are $92 \%$ confident that this species of bacteria takes, on average, between 7.249 hours and 7.773 hours to divide.

Solution

Solution Steps

Step 1: Calculate the Z-Score

To construct a $92\%$ confidence interval, we first determine the Z-Score corresponding to the confidence level. The Z-Score for a $92\%$ confidence level is approximately $Z = 1.7507$.

Step 2: Calculate the Margin of Error

The margin of error $E$ is calculated using the formula:

\[ E = Z \cdot \frac{s}{\sqrt{n}} \]

Substituting the values:

\[ E = 1.7507 \cdot \frac{0.8755}{\sqrt{34}} \approx 0.2629 \]

Step 3: Construct the Confidence Interval

The confidence interval for the mean is given by:

\[ \bar{x} \pm E \]

Substituting the sample mean $\bar{x} = 7.511$ and the margin of error $E = 0.2629$:

\[ 7.511 \pm 0.2629 \]

Calculating the lower and upper bounds:

\[ \text{Lower Bound} = 7.511 - 0.2629 \approx 7.248 \] \[ \text{Upper Bound} = 7.511 + 0.2629 \approx 7.774 \]