Statistic Help, and I need it fast

Question

A local radio station plays 40 rock-and-roll songs during each 4-hour show. The program director at the station needs to know the total amount of airtime for the 40 songs so that time can also be programmed during the show for news and advertisements. The distribution of the lengths of rock-and-roll songs, in minutes, is roughly symmetric with a mean length of 3.9 minutes and a standard deviation of 1.1 minutes.

a)Describe the sampling distribution of the sample mean song lengths for random samples of 40 rock-and-roll songs.

b)If the program manager schedules 80 minutes of news and advertisements for the 4-hour (240-minute) show, only 160 minutes are available for music. Approximately what is the probability that the total amount of time needed to play 40 randomly selected rock-and-roll songs exceeds the available airtime?

Answer 1

Well, let's right down what we know:

Part a): The distribution 'X' is symmetric, with a mean length of 3.9 minutes and a standard deviation of 1.1 minutes. That sounds like a Normal distribution to me. You're given the mean & standard deviation, so that's all you need to describe the distribution. The format is usually something like: N ( μ , σ ) You should be able to finish it from here, as I'm not allowed to do your homework for you.

Part b): Well, we figured out it's a normally distributed variable, that means we'll use a Z-test to test any hypothesis we're interested in. For this problem we're told that the program length is 240 minutes, but only 160 minutes are available for music. We're being asked to find the probability, that 40 randomly selected songs exceeds 160 minutes. Let's set up our null hypotheses: H₀ :(X>160) is what we need to test for. The alternative Hypothesis is: H_a :(X≤ 160), or that the 40 songs will NOT exceed the air time we have.

The easiest way would be to convert our normal distribution to a Standard Normal Distribution. To calculate a z-score, you'll need the data point you're interested in, the mean of the data set, and the standard deviation of the data set. The formula is: z = (x - μ) / σ, where x is your data point, μ is the mean, and σ is the standard deviation, you have all those numbers. That will give you a numeric value that you'll look up in the table for a std. normal dataset. You'll likely find them in the back of your textbook. Once again, I can't do the work for you, but I'm confident you can plug in the numbers, and look up the table value on your own.

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