a recent survey of 8 randomly selected social networking sites has a mean of 13.1 million visitors for a specific month. the standard deviation is 4.1 million find the confidence interval of the true mean
The confidence interval of the true mean is (9.0 million, 17.2 million).
To find the confidence interval of the true mean, we need to know the sample size, the sample mean, and the standard deviation. From the given information, we have a sample size of 8, a sample mean of 13.1 million visitors, and a standard deviation of 4.1 million.
The confidence interval can be calculated using the formula:
Confidence Interval = Sample Mean ± (Critical Value × Standard Error)
Step 1: Calculate the Standard Error:
Standard Error = Standard Deviation / √(Sample Size)
Plugging in the values:
Standard Error = 4.1 million / √8 ≈ 1.45 million
Step 2: Determine the Critical Value:
The critical value is based on the desired level of confidence and the degrees of freedom. Since the sample size is small (less than 30), we'll use a t-distribution instead of a z-distribution. Let's assume a 95% confidence level, which corresponds to a significance level (α) of 0.05.
The degrees of freedom for this sample is (n - 1) = (8 - 1) = 7.
Using a t-distribution table or a statistical calculator, the critical value for a 95% confidence level with 7 degrees of freedom is approximately 2.365.
Step 3: Calculate the Confidence Interval:
Confidence Interval = Sample Mean ± (Critical Value × Standard Error)
= 13.1 million ± (2.365 × 1.45 million)
Now, calculate the upper and lower bounds of the confidence interval:
Lower Bound = 13.1 million - (2.365 × 1.45 million)
≈ 9.83 million
Upper Bound = 13.1 million + (2.365 × 1.45 million)
≈ 16.37 million
So, the confidence interval of the true mean number of visitors to the social networking sites for a specific month is approximately 9.83 million to 16.37 million visitors.
To find the confidence interval of the true mean, we will use the formula:
Confidence Interval = Sample Mean ± (Critical Value * Standard Error)
Where Sample Mean is the mean of the 8 randomly selected social networking sites (13.1 million visitors), Critical Value represents the level of confidence, and Standard Error is the standard deviation divided by the square root of the sample size.
Let's assume we want a 95% confidence interval, which translates to a significance level (α) of 0.05. The critical value associated with this confidence level is ±1.96 (which can be found using a Z-table or a calculator).
Now let's calculate the confidence interval step-by-step:
1. Sample Mean = 13.1 million visitors
2. Standard Deviation (σ) = 4.1 million
3. Sample Size (n) = 8
4. Standard Error = σ / √n = 4.1 million / √8
(Note: √8 is approximately 2.83, so Standard Error ≈ 1.45 million)
5. Critical Value = 1.96 (for a 95% confidence level)
Confidence Interval = 13.1 ± (1.96 * 1.45)
Now let's calculate the lower and upper bounds of the confidence interval:
Lower Bound = 13.1 - (1.96 * 1.45)
Upper Bound = 13.1 + (1.96 * 1.45)
Lower Bound ≈ 10.25 million visitors
Upper Bound ≈ 15.95 million visitors
Therefore, the 95% confidence interval for the true mean number of visitors in a specific month is approximately 10.25 million to 15.95 million.