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Alternative Hypothesis on Credit Card Benefits

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The Thompson's Discount Appliance Store issues its own credit card. The credit manager wants to find whether the mean monthly unpaid balance is more than $400...
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The Thompson's Discount Appliance Store issues its own credit card. The credit manager wants to find whether the mean monthly unpaid balance
is more than $400. The level of significance is set at .05. A random check of 172 unpaid balances revealed the sample mean is $407 and the standard deviation of the sample is $38. Should the credit manager conclude the population mean is greater than $400, or is it reasonable that the difference of $7 ($407 - $400 =$7) is due to chance? ?

The null hypothesis Ho: the mean monthly unpaid balance is equal to $400.00.
The alternative hypothesis Ha: the mean monthly unpaid balance is more than $400.00.

Critical value z(.05)=1.645, rejection region is z > 1.645.
z-score is z=sqrt(172)*(407-400)/38=2.416.
2.416 > 1.645, we should reject Ho, thus the mean monthly unpaid balance is more than $400.00 at the level of .05.
It is unreasonable that the difference of $7.00 ($407-$400 = $7) is due to chance.

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The Thompson's Discount Appliance Store issues its own credit card. The credit manager wants to find whether the mean monthly unpaid balance is more than $400. The level of significance is set at .05. A random check of 172 unpaid balances revealed the sample mean is $407 and the standard deviation of the sample is $38. Should the credit manager conclude the population mean is greater than $400, or is it reasonable that the difference of $7 ($407 - $400 =$7) is due to chance? ?

The null hypothesis Ho: the mean monthly unpaid balance is equal to $400.00.
The alternative hypothesis Ha: the mean monthly unpaid balance is more than $400.00.

Crutical value z(.05)=1.645, rejection region is z > 1.645.
z-score is z=sqrt(172)*(407-400)/38=2.416.
2.416 > 1.645, we should reject Ho, thus the mean monthly unpaid balance is more than $400.00 at the level of .05.
It is unreasonable that the difference of $7.00 ($407-$400 = $7) is due to chance.

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The detailed explanation of the solution is attached I included the explanation of how to apply the Central Limit Theorem (if necessary), the verbal and graphical explanation of the decision rule.

The Thompson's Discount Appliance Store issues its own credit card. The credit manager wants to find whether the mean monthly unpaid balance
is more than $400. The level of significance is set at .05. A random check of 172 unpaid balances revealed the sample mean is $407 and the standard deviation of the sample is $38. ...

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