Question

A research scientist wants to know how many times per hour a certain strand of bacteria reproduces. The mean is found to be 11.4 reproductions and the population standard deviation is known to be 2. If a sample of 375 was used for the study, construct the 90% confidence interval for the true mean number of reproductions per hour for the bacteria. Round your answers to one decimal place.

Answer 1

Answer:

The 90% confidence interval for the true mean number of reproductions per hour for the bacteria is between 11.2 and 11.6 reproductions.

Step-by-step explanation:

We have that to find our $\alpha$ level, that is the subtraction of 1 by the confidence interval divided by 2. So:

$\alpha = \frac{1-0.9}{2} = 0.05$

Now, we have to find z in the Ztable as such z has a pvalue of $1-\alpha$.

So it is z with a pvalue of $1-0.05 = 0.95$, so $z = 1.645$

Now, find M as such

$M = z*\frac{\sigma}{\sqrt{n}}$

In which $\sigma$ is the standard deviation of the population and n is the size of the sample.

$M = 1.645*\frac{2}{\sqrt{375}} = 0.2$

The lower end of the interval is the sample mean subtracted by M. So it is 11.4 - 0.2 = 11.2 reproductions.

The upper end of the interval is the sample mean added to M. So it is 11.4 + 0.2 = 11.6 reproductions.

The 90% confidence interval for the true mean number of reproductions per hour for the bacteria is between 11.2 and 11.6 reproductions.

Answer 2

Answer:

90% confidence interval for the true mean number of reproductions per hour for the bacteria is between a lower limit of 11.2 and an upper limit of 11.6.

Step-by-step explanation:

Confidence interval for the true mean is given as mean +/- margin of error

mean = 11.4

sd = 2

n = 375

degree of freedom = n - 1 = 375 - 1 = 374

confidence level (C) = 90% = 0.9

significance level = 1 - C = 1 - 0.9 = 0.1 = 2%

critical value corresponding to 374 degrees of freedom and 10% significance level is 1.64926

E = critical value × sd/√n = 1.64926×2/√375 = 0.2

Lower limit of mean = mean - E = 11.4 - 0.2 = 11.2

Upper limit of mean = mean + E = 11.4 + 0.2 = 11.6

90% confidence interval is (11.2, 11.6)