Consider the average available for all hotel rooms from the previous assignment. Identify the point estimate of the average number of available hotel rooms in this lesson from each sample. Construct and Interpret a 95% confidence level for the true mean number of available hotel rooms, based on the point estimate of each sample. Describe how the confidence intervals will change depending on whether the population standard deviation is known or unknown.
To find the point estimate of the average number of available hotel rooms in each sample, you would need to calculate the sample mean for each sample. The formula for the sample mean is:
Sample Mean = Sum of all values in the sample / Number of values in the sample
After calculating the sample mean for each sample, you will have a point estimate for the average number of available hotel rooms in each sample.
To construct a 95% confidence interval for the true mean number of available hotel rooms, you would need to use the following formula:
Confidence Interval = Sample Mean ± (Critical Value * Standard Error)
The critical value is based on the desired confidence level (in this case, 95%). For a large sample size (usually considered to be above 30), you can use the z-score corresponding to the desired confidence level, which is approximately 1.96 for a 95% confidence level.
The standard error is calculated using the formula:
Standard Error = Sample Standard Deviation / Square Root of Sample Size
Now, regarding the difference in confidence intervals depending on whether the population standard deviation is known or unknown:
1. If the population standard deviation is known, you would use it in the standard error calculation. This is called a z-confidence interval. In this case, you would not need to estimate the standard deviation since you already know its value. The formula for the standard error remains the same, but you use the population standard deviation in the calculation.
2. If the population standard deviation is unknown, you would use the sample standard deviation in the standard error calculation. This is called a t-confidence interval. In this case, you estimate the population standard deviation using the sample standard deviation. The formula for the standard error remains the same, but you use the sample standard deviation in the calculation.
When the population standard deviation is known, confidence intervals tend to be narrower because there is less uncertainty about the population parameter. On the other hand, when the population standard deviation is unknown and a sample standard deviation is used instead, confidence intervals tend to be wider because there is more uncertainty about the population parameter.