Bacterial samples were taken from several surfaces in a daycare and incubated in petri dishes. After that the number of bateria colonies were counted, with the following results 412, 437, 425, 476, 451, 483.
1) construct a 95% confidence interval for the mean number of colonies
2) after constructing the interval above, the resarcher notices that it is quite wide, and wishes to do a follow-up study with greater precision. How large a sample is reuired to produce a 95% confidence interval with margin of error 5?
IM SO CONFUSED!!
Since your sample size is small and the problem doesn't say that the population has a normal distribution, you may want to use a t-table to determine the 95% confidence interval. A general example of a formula is: CI95 = mean + or - (t-value)(sd/√n)
Note: sd = standard deviation.
You will need to determine mean and standard deviation before you can plug the values into the formula to determine the interval. The sample size is 6.
For the second part, you will need a formula to determine sample size. There are different formulas you can use; here is an example: n = [(t-value) * sd)/E]^2
Note: E = 5
I hope this will help.
One other comment: if you are expected to use a z-value instead of a t-value, the z-value would be 1.96 using a z-table for a normal distribution.
I understand that you're feeling confused about the questions related to constructing a confidence interval for the mean number of colonies and determining the sample size for a desired margin of error. Let me break it down for you step by step.
1) Constructing a 95% confidence interval for the mean number of colonies:
To construct a confidence interval, we need to have the sample mean, standard deviation, and sample size. From the information you provided, we have the following number of colonies: 412, 437, 425, 476, 451, and 483.
To find the sample mean, we calculate the average of these numbers:
(412 + 437 + 425 + 476 + 451 + 483) / 6 = 428.33 (approximately)
Next, we need to calculate the standard deviation of the sample. For simplicity, I'll assume this is the sample standard deviation.
Using the formula for sample standard deviation, we find the sum of the squared differences between each number and the sample mean:
Sum = (412 - 428.33)^2 + (437 - 428.33)^2 + (425 - 428.33)^2 + (476 - 428.33)^2 + (451 - 428.33)^2 + (483 - 428.33)^2
Then, divide the sum by the sample size minus 1 (in this case 6-1=5):
Standard deviation = square root(Sum / (sample size - 1))
After calculating the standard deviation, we can construct the confidence interval.
The formula for a confidence interval for the mean, assuming a normal distribution, is:
Confidence interval = sample mean ± (t * (standard deviation / √n))
Where:
- t is the critical value calculated from the t-distribution table for the desired level of confidence (95% in this case). The number of degrees of freedom is n - 1 (sample size minus 1).
- √n is the square root of the sample size.
Using statistical software or a t-distribution table, you can find the corresponding value for t. Let's assume it is 2.571 for the given sample size of 6.
Substituting the values into the formula, we get:
Confidence interval = 428.33 ± (2.571 * (standard deviation / √6))
Now, you can calculate the upper and lower bounds of the confidence interval.
2) Determining the sample size for a desired margin of error:
To determine the sample size required for a desired margin of error, you need to use the formula:
Sample size = (z^2 * (standard deviation^2)) / (margin of error^2)
Where:
- z is the critical value from the standard normal distribution table corresponding to the desired level of confidence. For a 95% confidence level, z is approximately 1.96.
- standard deviation is the estimated standard deviation of the population (if unknown, you can use the sample standard deviation).
- margin of error is the maximum allowable difference between the sample mean and the population mean.
Using the given margin of error of 5 and the estimated standard deviation from the previous question, you can calculate the required sample size:
Sample size = (1.96^2 * (standard deviation^2)) / (5^2)
I hope this explanation helps clarify the process of constructing a confidence interval and determining the sample size. Remember, these calculations can be complex, and using statistical software or consulting with a statistician can provide more accurate results.