Question

in a sample of 41 u.s. adults aged 18 - 24 who celebrate halloween, the mean amount spent on a costume was $38.8 with a standard deviation of $13.15. round answers to at least 4 decimal places.
a) what parameter are we estimating?
b) we can work this problem because:
c) what is the point estimate of that parameter?
d) what is the standard deviation of the sampling distribution for the sample means?
e) find the margin of error for a 95% confidence interval for the true average amount spent on a costume.
f) we are 95% confident that the true average amount spent on a halloween costume for u.s. adults aged 18 - 24 is between $ and $. do not round between steps, and round answer to 4 decimal places.

Explanation:

Step1: Recall the formula for margin of error

The formula for margin of error \(E = z_{\alpha/2}\times\frac{\sigma}{\sqrt{n}}\). For a 95% confidence interval, \(\alpha=1 - 0.95=0.05\), so \(\alpha/2=0.025\). From the standard normal distribution table, \(z_{0.025} = 1.96\). We know that \(\sigma = 13.15\) and \(n = 41\).

Step2: Calculate the margin of error

Substitute the values into the formula: \(E=1.96\times\frac{13.15}{\sqrt{41}}\). First, calculate \(\sqrt{41}\approx6.4031\). Then \(\frac{13.15}{6.4031}\approx2.0537\). Finally, \(E = 1.96\times2.0537=4.0253\).

Step3: Calculate the confidence interval

The confidence interval is \(\bar{x}-E<\mu<\bar{x} + E\). Given \(\bar{x}=38.8\), \(\bar{x}-E=38.8 - 4.0253=34.7747\) and \(\bar{x}+E=38.8+4.0253 = 42.8253\).

Answer:

e) \(4.0253\)
f) \(34.7747\) and \(42.8253\)