Final answer:
To approximate the 95% confidence interval for the population mean using the 2 SD method, we calculate the margin of error by multiplying the standard deviation by 2 and add/subtract it from the sample mean. The interval from question 5 has a narrower width compared to the interval calculated with a larger standard deviation. The changes in the interval make sense intuitively because a larger standard deviation leads to a wider range of plausible values for the population mean.
Step-by-step explanation:
To approximate the 95% confidence interval for the population mean, we will use the 2 SD method. Given the sample standard deviation (s) of $300, we can calculate the margin of error by multiplying the standard deviation by 2, which gives us $600. To find the lower and upper bounds of the interval, we subtract and add the margin of error from the sample mean. So the lower bound is $942.5 - $600 = $342.5 and the upper bound is $942.5 + $600 = $1542.5.
Comparing this interval to the one from question 5, we can see that the midpoint (average planned spending) remains the same at $942.5. However, the width of the interval has increased from $942.5 - $755 = $187.5 to $1542.5 - $342.5 = $1200.
Intuitively, this change makes sense because a larger standard deviation implies more variability in the data, leading to a wider interval. In other words, with a larger value for s, the range of plausible values for the population mean increases.