118k views
3 votes
A travel agent is interested in the average price of a hotel room during the summer in a resort community. The agent randomly selects 18 hotels from the community and determines the price of a regular room with a king size bed. The average price of the room for the sample was $120 with a standard deviation of $35. Assume the prices are normally distributed. Construct an interval to estimate the true average price of a regular room with a king size bed in the resort community with 99% confidence. Round the endpoints to two decimal places, if necessary. Answer​

User Catsy
by
8.0k points

1 Answer

5 votes

Answer: Therefore, we can say with 99% confidence that the true average price of a regular room with a king size bed in the resort community is between $94.20 and $145.80.

Explanation:

To construct a confidence interval to estimate the true average price of a regular room with a king size bed in the resort community, we can use the formula: Confidence Interval = Sample Mean ± (Critical Value * Standard Error) 1. Calculate the critical value: Since we want a 99% confidence interval, we need to find the critical value that corresponds to a 99% confidence level and a sample size of 18. Using a t-distribution table or calculator, the critical value is approximately 2.898. 2. Calculate the standard error: The standard error represents the average amount of variation in the sample mean. It is calculated by dividing the standard deviation by the square root of the sample size. Standard Error = Standard Deviation / √(Sample Size) In this case, the standard deviation is $35 and the sample size is 18. Therefore, the standard error is $35 / √18 ≈ $8.24 (rounded to two decimal places). 3. Calculate the confidence interval: Using the formula, we can now calculate the confidence interval: Confidence Interval = Sample Mean ± (Critical Value * Standard Error) Plugging in the values: Confidence Interval = $120 ± (2.898 * $8.24) Now, let's calculate the confidence interval: Lower Limit = $120 - (2.898 * $8.24) Upper Limit = $120 + (2.898 * $8.24) Rounding to two decimal places, the lower limit is approximately $94.20 and the upper limit is approximately $145.80.

User Setop
by
8.6k points