T Distribution: Average Revenue Range for All Shops | International Chain

In summary, to estimate the average revenue of all shops owned by an international chain, a confidence interval can be calculated using the formula xbar +/- t*(s/√n). The Chi-square distribution may also be used to calculate a confidence interval for the standard deviation. The appropriate alpha value to use depends on the desired level of confidence.
  • #1
rhyno89
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Homework Statement


Revenues reported last week from nine shops owned by an international chain averaged $59,540 with a standard deviation of $6,860. Based on these figures in what range might the company expect to find the average revenue of all shops? Include all assumptions


Homework Equations





The Attempt at a Solution



I think i have the answer but it never hurts for a little clarification.

Anyway I started with establishing a sufficient alpha. Since it asks to include all shops and not say 95% of shops I used an alpha value of .01 and since its a two tail test that makes it .005. There are nine shops and therefore 8 df so the t stat is equivalent to t.005,8. From here it was simply xbar +/- t*(s/3)

My only question is if that alpha value will satisfy the "all stores" aspect and whether or not I should first attempt to find an estimate of the standard deviation usingt he choi square distribution and then to plug that into the confidence interval...thanks
 
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  • #2


Assuming that the data is normally distributed, you can use the t-distribution to calculate a confidence interval for the average revenue of all shops. However, since the sample size is small (n=9), the t-distribution may not be appropriate and the use of the Chi-square distribution may be more accurate.

To calculate the confidence interval, you can use the formula xbar +/- t*(s/√n), where xbar is the sample mean, s is the sample standard deviation, t is the critical value from the t-distribution or Chi-square distribution, and n is the sample size.

Since the sample standard deviation is given as $6,860, you can use this value in the formula. However, if you want to calculate an estimate of the population standard deviation, you can use the Chi-square distribution to calculate a confidence interval for the standard deviation. Once you have this estimate, you can use it in the formula for the confidence interval for the average revenue.

As for the alpha value, it depends on the level of confidence you want to use. If you want to be 99% confident in your results, then using an alpha of .01 is appropriate. However, if you want to be 95% confident, then you should use an alpha of .05. This will give you a wider confidence interval, but it will also provide a more conservative estimate.

In conclusion, to calculate a confidence interval for the average revenue of all shops, you can use the formula xbar +/- t*(s/√n), where xbar is the sample mean, s is the sample standard deviation, t is the critical value from the t-distribution or Chi-square distribution, and n is the sample size. You can also use the Chi-square distribution to calculate a confidence interval for the standard deviation, if desired. The alpha value you use will depend on the level of confidence you want to have in your results.
 

What is a T distribution and when is it used?

A T distribution, also known as a Student's T distribution, is a probability distribution that is used to estimate the mean of a population when the sample size is small and the population standard deviation is unknown. It is often used in hypothesis testing and confidence interval calculations.

How is a T distribution different from a normal distribution?

A T distribution is similar to a normal distribution in shape, but it has heavier tails, meaning there is a higher probability of extreme values. This makes it more suitable for small sample sizes where the population standard deviation is unknown.

What are the key characteristics of a T distribution?

The key characteristics of a T distribution are its degree of freedom, center, and spread. The degree of freedom is equal to the sample size minus one. The center of the distribution is at zero, and the spread is determined by the degree of freedom.

How do you interpret the degrees of freedom in a T distribution?

The degrees of freedom in a T distribution represent the number of independent pieces of information used to estimate a population parameter. In other words, it is the number of values in a sample that are free to vary when estimating a population parameter.

What is the relationship between a T distribution and a Z distribution?

A T distribution approaches a Z distribution as the sample size increases. This means that for large sample sizes, the T distribution is very similar to a Z distribution. However, for small sample sizes, the T distribution is wider and has heavier tails than a Z distribution.

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