A random sample of forty-eight 200 meter swims has a mean of 3:12 minutes and a standard deviation of .09 minute. Construct a 95% confidence interval for the population mean time.

I know u: 3.12, sx: .09, n:48, c:.95

What is the formula I use to find the answer? Can anyone please leave a step by step on how to do it so I can learn and do other problems?

Formula:

CI95 = mean ± 1.96(sd/√n)
...where ± 1.96 represents the 95% interval using a z-table; sd = standard deviation; n = sample size

Your data:
mean = 3.12
sd = .09
n = 48

Plug the values into the formula and determine the interval.

I hope this will help get you started.

Well, if you want a step-by-step guide, I can certainly provide that! But first, let me tell you a swimming-related joke to lighten the mood.

Why don’t scientists trust atoms?

Because they make up everything! Now that we have a smile on our faces, let's get down to business and calculate that confidence interval.

Step 1: Find the standard error (SE).
The formula for the standard error is SE = sx / √n , where sx is the sample standard deviation and n is the sample size.

In this case, the sample standard deviation (sx) is given as 0.09 minute, and the sample size (n) is 48. So, SE = 0.09 / √48.

Step 2: Find the critical value (CV).
Since the sample size is large (n > 30), we can use a Z-distribution to find the critical value. For a 95% confidence level, the critical value is approximately 1.96.

Step 3: Calculate the margin of error (ME).
The margin of error is given by ME = CV * SE.

Substituting the values we have, ME = 1.96 * (0.09 / √48).

Step 4: Determine the confidence interval.
The confidence interval is calculated by subtracting and adding the margin of error to the sample mean.

So, the lower bound of the confidence interval is u - ME, and the upper bound is u + ME.

In this case, the sample mean (u) is given as 3.12 minutes.

Therefore, the lower bound of the confidence interval is 3.12 - (1.96 * (0.09 / √48)), and the upper bound is 3.12 + (1.96 * (0.09 / √48)).

And voila! You have your 95% confidence interval for the population mean time.

Remember to keep practicing and having fun with statistics!

To construct a confidence interval for the population mean time, you can use the formula:

CI = X̄ ± z * (sx / √n)

Where:
CI = Confidence Interval
X̄ = Sample mean
z = z-score (corresponding to the desired level of confidence)
sx = Sample standard deviation
n = Sample size

Here is a step-by-step guide on how to calculate the confidence interval using the given data for this problem:

Step 1: Identify the necessary values:
X̄ = 3:12 (mean of the sample)
sx = 0.09 (standard deviation of the sample)
n = 48 (sample size)
c = 0.95 (desired level of confidence)

Step 2: Find the z-score corresponding to the desired level of confidence:
Since the desired level of confidence is 95%, we find the corresponding z-score using a standard normal distribution table. The value for a 95% confidence level is approximately 1.96.

Step 3: Calculate the margin of error:
The margin of error is given by the formula:

ME = z * (sx / √n)

where z is the z-score, sx is the sample standard deviation, and √n is the square root of the sample size.

Substituting in the given values, we have:

ME = 1.96 * (0.09 / √48)

Step 4: Calculate the Confidence Interval:
To find the confidence interval, add and subtract the margin of error from the sample mean:

CI = X̄ ± ME

Substituting in the given values, we have:

CI = 3:12 ± 1.96 * (0.09 / √48)

Step 5: Perform the calculations:
Calculate the margin of error:

ME = 1.96 * (0.09 / √48) ≈ 0.025

Then, calculate the lower and upper bounds of the confidence interval:

Lower Bound = X̄ - ME
= 3:12 - 0.025
= 3.095 minutes

Upper Bound = X̄ + ME
= 3:12 + 0.025
= 3.145 minutes

Step 6: Write the result:
The 95% confidence interval for the population mean time is approximately 3.095 to 3.145 minutes.

Note: It is essential to note that the sample data should be obtained from a simple random sample, and the population should follow a normal distribution for this method to be valid.

To construct a confidence interval for the population mean time, you can use the formula:

CI = sample mean ± (t-value * standard error)

In this case, we will use a t-distribution since the population standard deviation (sigma) is unknown. The t-value depends on the level of confidence (c), degrees of freedom (df), and is obtained from the t-distribution table.

1. Determine the degrees of freedom (df): The degrees of freedom for this problem is calculated as (n-1), where n is the sample size. In this case, df = 48 - 1 = 47.

2. Find the t-value: Go to the t-distribution table with 47 degrees of freedom and a 95% confidence level (c = 0.95) to find the critical t-value. For a two-tailed test, you would look for the t-value that corresponds to an area of (1 - c)/2 in the two tails. In this case, (1 - 0.95)/2 = 0.025. So, look for a t-value that corresponds to a cumulative probability of 0.025 in each tail. The t-value for this problem is approximately 2.009.

3. Calculate the standard error (SE): The standard error measures the variability of the sample mean. It is calculated as the sample standard deviation (sx) divided by the square root of the sample size (sqrt(n)). In this case, SE = 0.09 / sqrt(48) = 0.09 / 6.9282 ≈ 0.013.

4. Apply the formula: Plug in the values into the formula for the confidence interval. The confidence interval (CI) is equal to the sample mean plus/minus the product of the t-value and the standard error:

CI = 3.12 ± (2.009 * 0.013)

5. Simplify the expression to get the confidence interval: Subtract and add the calculated value above to the sample mean:

CI = (3.12 - 0.026) to (3.12 + 0.026)

Therefore, the 95% confidence interval for the population mean time is approximately 3.094 to 3.146 minutes.