Question

question 8
in an examination of purchasing patterns of shoppers, a sample of 25 shoppers revealed that they spent
on average, $50 per hour of shopping. the sample standard deviation is $4.80 per hour of shopping.
assuming that the amount spent per hour of shopping is normally distributed, calculate a 90% confidence
interval.
point estimate =
alpha =
critical value =
standard error =
margin of error = +-
confidence interval lower limit =
confidence interval upper limit =

Explanation:

Step1: Determine the point estimate

The point estimate for the population mean is the sample mean. Here, the sample mean is given as $\$50$.
Point estimate = $50$

Step2: Calculate $\alpha$

The confidence level is $90\%$. So, $\alpha=1 - 0.90=0.10$

Step3: Find the critical value

Since the sample size $n = 25$ (small sample, $n<30$) and the population standard deviation is unknown (we use the sample standard deviation $s = 4.80$), we use the $t$-distribution.
The degrees of freedom $df=n - 1=25-1 = 24$.
For $\alpha=0.10$ and $df = 24$, using the $t$-table or a calculator, the critical value $t_{\alpha/2}=t_{0.05,24}=1.711$

Step4: Calculate the standard error

The formula for the standard error $SE=\frac{s}{\sqrt{n}}$.
Substituting $s = 4.80$ and $n = 25$, we get $SE=\frac{4.80}{\sqrt{25}}=\frac{4.80}{5}=0.96$

Step5: Calculate the margin of error

The margin of error $E=t_{\alpha/2}\times SE$.
Substituting $t_{\alpha/2}=1.711$ and $SE = 0.96$, we get $E=1.711\times0.96 = 1.64256\approx1.64$

Step6: Calculate the confidence interval limits

The lower limit of the confidence interval $= \bar{x}-E$.
Substituting $\bar{x}=50$ and $E = 1.64$, we get $50-1.64 = 48.36$
The upper limit of the confidence interval $=\bar{x}+E$.
Substituting $\bar{x}=50$ and $E = 1.64$, we get $50 + 1.64=51.64$

Answer:

Point estimate = $50$
Alpha = $0.10$
Critical Value = $1.711$
Standard error = $0.96$
Margin of error = $\pm1.64$
Confidence interval lower limit = $48.36$
Confidence interval upper limit = $51.64$