Question

a random sample of 10 parking meters in a beach community showed the following incomes for a day. assume the incomes are normally distributed. $3.60, $4.50, $2.80, $6.30, $2.60, $5.20, $6.75, $4.25, $8.00, $3.00 find the 95% confidence interval for the true mean. a. ($3.39,$6.01) b. ($2.11,$5.34) c. ($1.35,$2.85) d. ($4.81,$6.31)

Explanation:

Step1: Calculate the sample mean

The formula for the sample mean $\bar{x}$ is $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$.
Here, $n = 10$, and $x_{i}$ are the data points: $3.60,4.50,2.80,6.30,2.60,5.20,6.75,4.25,8.00,3.00$.
$\sum_{i=1}^{10}x_{i}=3.60 + 4.50+2.80 + 6.30+2.60+5.20+6.75+4.25+8.00+3.00=47$.
$\bar{x}=\frac{47}{10}=4.7$.

Step2: Calculate the sample standard deviation

The formula for the sample standard deviation $s$ is $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$.
$(x_{1}-\bar{x})^{2}=(3.60 - 4.7)^{2}=(-1.1)^{2}=1.21$.
$(x_{2}-\bar{x})^{2}=(4.50 - 4.7)^{2}=(-0.2)^{2}=0.04$.
$(x_{3}-\bar{x})^{2}=(2.80 - 4.7)^{2}=(-1.9)^{2}=3.61$.
$(x_{4}-\bar{x})^{2}=(6.30 - 4.7)^{2}=(1.6)^{2}=2.56$.
$(x_{5}-\bar{x})^{2}=(2.60 - 4.7)^{2}=(-2.1)^{2}=4.41$.
$(x_{6}-\bar{x})^{2}=(5.20 - 4.7)^{2}=(0.5)^{2}=0.25$.
$(x_{7}-\bar{x})^{2}=(6.75 - 4.7)^{2}=(2.05)^{2}=4.2025$.
$(x_{8}-\bar{x})^{2}=(4.25 - 4.7)^{2}=(-0.45)^{2}=0.2025$.
$(x_{9}-\bar{x})^{2}=(8.00 - 4.7)^{2}=(3.3)^{2}=10.89$.
$(x_{10}-\bar{x})^{2}=(3.00 - 4.7)^{2}=(-1.7)^{2}=2.89$.
$\sum_{i = 1}^{10}(x_{i}-\bar{x})^{2}=1.21+0.04 + 3.61+2.56+4.41+0.25+4.2025+0.2025+10.89+2.89=30.465$.
$s=\sqrt{\frac{30.465}{9}}\approx1.84$.

Step3: Determine the critical value

For a 95% confidence interval and $n-1 = 9$ degrees of freedom, the critical value $t_{\alpha/2}$ from the t - distribution table is $t_{0.025,9}=2.262$.

Step4: Calculate the margin of error

The margin of error $E=t_{\alpha/2}\frac{s}{\sqrt{n}}$.
$E = 2.262\times\frac{1.84}{\sqrt{10}}\approx2.262\times0.582\approx1.31$.

Step5: Calculate the confidence interval

The confidence interval is $\bar{x}-E<\mu<\bar{x}+E$.
$4.7-1.31 = 3.39$ and $4.7 + 1.31=6.01$.

Answer:

A. ($3.39,$6.01)