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Marysya12 [62]
3 years ago
7

Statistics In the manual “How to Have a Number One the Easy Way,” it is stated that a song “must be no longer than three minutes

and thirty seconds (210 seconds)”. A simple random sample of 40 current hit songs results in a mean length of 252.5 seconds. Assume that the standard deviation of song lengths is 54.5 sec. Use a 0.05 significance level to test the claim that the sample is from a population of songs with a mean greater than 210 seconds.
Mathematics
1 answer:
dimaraw [331]3 years ago
4 0

Answer:

We conclude that the sample is from a population of songs with a mean greater than 210 seconds.

Step-by-step explanation:

We are given that a simple random sample of 40 current hit songs results in a mean length of 252.5 seconds.

Assume that the standard deviation of song lengths is 54.5 sec.

Let \mu = <u><em>population mean length of the songs</em></u>

So, Null Hypothesis, H_0 : \mu \leq 210 seconds      {means that the sample is from a population of songs with a mean smaller than or equal to 210 seconds}

Alternate Hypothesis, H_A : \mu > 210 seconds      {means that the sample is from a population of songs with a mean greater than 210 seconds}

The test statistics that will be used here is <u>One-sample z-test statistics </u>because we know about population standard deviation;

                              T.S.  =  \frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }  ~ N(0,1)

where, \bar X = sample mean length of songs = 252.5 seconds

            \sigma = population standard deviation = 54.5 seconds

            n = sample of current hit songs = 40

So, <u><em>the test statistics</em></u> =  \frac{252.5-210}{\frac{54.5}{\sqrt{40} } }

                                   =  4.932

The value of z-test statistics is 4.932.

Now, at 0.05 level of significance, the z table gives a critical value of 1.645 for the right-tailed test.

Since the value of our test statistics is more than the critical value of z as 4.932 > 1.645, so <u><em>we have sufficient evidence to reject our null hypothesis</em></u> as it will fall in the rejection region.

Therefore, we conclude that the sample is from a population of songs with a mean greater than 210 seconds.

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