the times a musician spends performing a rock song and a country song are approximately normally…

the times a musician spends performing a rock song and a country song are approximately normally distributed. the rock song has a mean time of 133 seconds with a standard deviation of 3.5 seconds, and the country song has a mean time of 126 seconds with a standard deviation of 3.3 seconds. if the musician randomly selects 4 times the rock song is played and 3 times the country song is played, what is the probability that the mean time for the rock song is less than the mean time for the country song?\n0.0034\n0.0365\n0.9964\n0.9966

the times a musician spends performing a rock song and a country song are approximately normally distributed. the rock song has a mean time of 133 seconds with a standard deviation of 3.5 seconds, and the country song has a mean time of 126 seconds with a standard deviation of 3.3 seconds. if the musician randomly selects 4 times the rock song is played and 3 times the country song is played, what is the probability that the mean time for the rock song is less than the mean time for the country song?\n0.0034\n0.0365\n0.9964\n0.9966

Answer

Explanation:

Step1: Define variables

Let $\bar{X}_1$ be the sample - mean of rock - song times and $\bar{X}_2$ be the sample - mean of country - song times. $\mu_1 = 133$, $\sigma_1 = 3.5$, $n_1 = 4$, $\mu_2 = 126$, $\sigma_2 = 3.3$, $n_2 = 3$.

Step2: Calculate the mean and standard deviation of $\bar{X}_1-\bar{X}_2$

The mean of $\bar{X}_1-\bar{X}_2$ is $\mu=\mu_1 - \mu_2=133 - 126 = 7$. The standard deviation of $\bar{X}_1-\bar{X}_2$ is $\sigma=\sqrt{\frac{\sigma_1^{2}}{n_1}+\frac{\sigma_2^{2}}{n_2}}=\sqrt{\frac{3.5^{2}}{4}+\frac{3.3^{2}}{3}}=\sqrt{\frac{12.25}{4}+\frac{10.89}{3}}=\sqrt{3.0625 + 3.63}=\sqrt{6.6925}\approx2.587$.

Step3: Standardize the random variable

We want to find $P(\bar{X}_1<\bar{X}_2)$, which is equivalent to $P(\bar{X}_1 - \bar{X}_2<0)$. Let $Z=\frac{(\bar{X}_1 - \bar{X}_2)-\mu}{\sigma}$. Substituting the values, we get $Z=\frac{0 - 7}{2.587}\approx - 2.71$.

Step4: Find the probability

Using the standard normal distribution table, $P(Z < - 2.71)=0.0034$.

Answer:

0.0034