The average price for new mobile homes is known to be distributed with a standard deviation of 350. if a random rample of 49 homes resulted in a sample mean of 38500, obtain a 90% confidence interval on the population mean.
To obtain a confidence interval for the population mean, we can use the formula:
Confidence interval = sample mean ± (critical value * standard deviation / √sample size)
Since the population standard deviation (σ) is known to be 350, we can directly use it in our calculations.
Step 1: Find the critical value
Since we want a 90% confidence interval, we need to find the critical value for a 90% confidence level. This critical value corresponds to the z-score needed to capture 90% of the area under the normal distribution curve. We can find this value using a standard normal distribution table or a calculator.
For a 90% confidence level, the critical value is approximately 1.645.
Step 2: Calculate the standard error
The standard error (SE) is the standard deviation divided by the square root of the sample size (√n).
SE = σ / √n
In this case, the sample size is 49 and the standard deviation is given as 350. So, the standard error is:
SE = 350 / √49 = 350 / 7 = 50
Step 3: Calculate the confidence interval
Now, we can substitute the values into the confidence interval formula:
Confidence interval = sample mean ± (critical value * standard error)
Confidence interval = 38500 ± (1.645 * 50)
Confidence interval = 38500 ± 82.25
Confidence interval = (38500 - 82.25, 38500 + 82.25)
Confidence interval = (38417.75, 38582.25)
Therefore, the 90% confidence interval for the population mean is (38417.75, 38582.25).