Final answer:
To calculate the 95% confidence interval for the mean amount students spend per semester, we use the z-distribution formula with a z-score of 1.96, given the known population standard deviation. The calculated confidence interval is approximately $341 to $367.
Step-by-step explanation:
To calculate the 95% confidence interval for the population mean when the population standard deviation is known, we use the z-distribution. The formula for the confidence interval is:
Mean ± (z-value) * (Population Standard Deviation / sqrt(n))
Where Mean is the sample mean, z-value is the z-score corresponding to the 95% confidence level (which is 1.96 for two-tailed tests), and n is the sample size. In this case:
- Mean = $354
- Population Standard Deviation = $45
- n = 46
- z-value = 1.96
Calculating the margin of error:
Margin of error = 1.96 * ($45 / sqrt(46))
= 1.96 * ($45 / 6.7823)
= 1.96 * 6.6354
= approximately $13.00
The 95% confidence interval is then:
$354 ± $13.00
= ($341, $367)
Therefore, we can be 95% confident that the population mean for the amount students spend per semester is between $341 and $367.