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The manager of a customer support center takes an SRS of 40 reported issues and finds that 22% of the sampled issues required more than one call to resolve. The manager may take an SRS like this each month. Suppose that it is really an average of 25% of the approximately 1000 issues reported per month that require more than one call.


Let represent the proportion of a sample of 40 reported issues that require more than one call to resolve. What are the mean and standard deviation of the sampling distribution of p?

Sagot :

Answer:

The mean of the sampling distribution of p is 0.25 and the standard deviation is 0.0685.

Step-by-step explanation:

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]

25% of the approximately 1000 issues reported per month that require more than one call.

This means that [tex]p = 0.25[/tex]

What are the mean and standard deviation of the sampling distribution of p?

Sample of 40 means that [tex]n = 40[/tex].

By the Central Limit Theorem,

The mean is [tex]\mu = p = 0.25[/tex]

The standard deviation is [tex]s = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.25*0.75}{40}} = 0.0685[/tex]

The mean of the sampling distribution of p is 0.25 and the standard deviation is 0.0685.