Question

sony would like to test the hypothesis that the average age of a playstation user is different from the average age of an xbox user. a random sample of 36 playstation users had an average age of 34.2 years while a random sample of 30 xbox users had an average age of 32.7 years. assume that the population standard deviation for the age of playstation and xbox users is 3.9 and 4.0 years, respectively. sony would like to set α = 0.10. what is the 90% confidence interval for the difference in population means? round to two decimal places as needed.

a. (1.18, 1.82)
b. (-0.11, 3.11)
c. (0.65, 2.35)
d. (-1.70, 4.70)

Explanation:

Step1: Calculate the difference in sample means

The difference in sample means $\bar{x}_1-\bar{x}_2 = 34.2 - 32.7=1.5$

Step2: Calculate the standard error

The formula for the standard error $SE=\sqrt{\frac{s_1^{2}}{n_1}+\frac{s_2^{2}}{n_2}}$
where $s_1 = 3.9$, $n_1=36$, $s_2 = 4.0$, $n_2 = 30$
$SE=\sqrt{\frac{3.9^{2}}{36}+\frac{4.0^{2}}{30}}=\sqrt{\frac{15.21}{36}+\frac{16}{30}}=\sqrt{0.4225 + 0.5333}=\sqrt{0.9558}\approx0.98$

Step3: Find the critical value

For a 90% confidence interval, $\alpha=0.10$ and $\alpha/2 = 0.05$. Using the standard normal distribution (since sample sizes are large, $n_1 = 36\geq30$ and $n_2=30\geq30$), the critical value $z_{\alpha/2}=z_{0.05}\approx1.645$

Step4: Calculate the margin of error

Margin of error $E = z_{\alpha/2}\times SE=1.645\times0.98\approx1.61$

Step5: Calculate the confidence interval

The confidence interval is $(\bar{x}_1-\bar{x}_2 - E,\bar{x}_1-\bar{x}_2 + E)$
$=(1.5 - 1.61,1.5+1.61)=(- 0.11,3.11)$

Answer:

B. (-0.11, 3.11)