Customers are considered very satisfied with the XYZ Box video game system if the mean of the population of all satisfaction ratings is higher than 50. Assume that the standard deviation of the population of all satisfaction ratings is 4.64.If the population mean of all satisfaction ratings is 50, what is the probability of observing a random sample of 80 satisfaction ratings with a mean greater than or equal to 51.45?We have actually observed a random sample of 80 satisfaction ratings with a mean of 51.45. What do you conclude about whether customers are very satisfied with the XYZ Box video game system?

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Asked Mar 31, 2019
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     Customers are considered very satisfied with the XYZ Box video game system if the mean of the population of all satisfaction ratings is higher than 50. Assume that the standard deviation of the population of all satisfaction ratings is 4.64.

  1. If the population mean of all satisfaction ratings is 50, what is the probability of observing a random sample of 80 satisfaction ratings with a mean greater than or equal to 51.45?
  2. We have actually observed a random sample of 80 satisfaction ratings with a mean of 51.45. What do you conclude about whether customers are very satisfied with the XYZ Box video game system?
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Expert Answer

Step 1

1.

Denote X as the mean of all the satisfaction ratings. It is given that X is distributed, with mean μ =50 and standard deviation σ = 4.64.

A random sample of 80 satisfaction ratings is taken.

As the sample size is large (>30), it can be assumed that the mean of all the satisfaction ratings follows normal distribution with mean μ = 50 and standard deviation σ = 4.64.

Let X bar be the mean of all the sample satisfaction ratings. It is known that,

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Step 2

The probability that the sample mean satisfaction rating is greater than or equal to 51.45 is,

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Step 3

2.

For a sample of 80 satisfaction ratings, the mean is 51.45 and the standard deviation is 4.64. It is claimed that the mean of the population of all satisfaction ratings is higher than 50. Consider the level of significance for the test as α=0.05.

The test hypotheses are given below:

Null hypothesis:

H0: µ ≤ 50

That is, the mean of the population of all satisfaction ratings is lower than 50.

Alternative hypothesis:

H1: µ > 50

That is, the mean of the population of all satisfaction ratings is higher than 50.

Critical value:

The critical value is obtained using the...

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