Question

A recent survey concluded that the proportion of american teenagers who have a cell phone is 0.27. the true population proportion of american teenagers who have a cell phone is 0.29. for samples of size 1,000 that are selected at random from this population, what are the mean and standard deviation, respectively, for the sampling distribution of the sample proportion of american teenagers who have a cell phone?

Answer 1

Answer:

The mean for the sampling distribution of the sample proportion is 0.29

The standard deviation for the sampling distribution of the sample proportion is 0.01435

Step-by-step explanation:

The mean for the sampling distribution of the sample proportion is always equal to the true population proportion, in this case; p = 0.29

The standard deviation for the sampling distribution of the sample proportion is calculated as;

$\sqrt{\frac{p(1-p)}{n} }$

Using the given values;

p = 0.29

1 - p = 0.71

n = 1000

The standard deviation becomes;

$\sqrt{\frac{(0.29)(1-0.29)}{1000} }$

The s.d becomes 0.01435

Answer 2

The mean and standard deviation for the sampling distribution of the sample proportion of American teenagers with a cell phone are:

• Mean = 0.29
• SD = 0.014

Recall:

• Sample distribution mean of a sample proportion = true population proportion.
• Formula for standard deviation of a sample distribution of a sample proportion is given as: $\mathbf{\sqrt{\frac{p(1-p)}{n} } }$

Given that:

Population proportion, p, is 0.29

Therefore,

• Mean for the sample distribution of the sample proportion = 0.29.

Standard Deviation for the sampling distribution of the sample proportion of American teenagers with a cell phone =  $\mathbf{\sqrt{\frac{p(1-p)}{n} } }$

• Where,

p = 0.29

n = 1,000

• Substitute

$SD = \sqrt{\frac{0.29(1-0.29)}{100} }\\\\SD = \sqrt{\frac{0.29(0.71)}{100}}\\\\SD = \sqrt{\frac{0.2059}{100}}\\\\\mathbf{SD = 0.014}$

Therefore, the mean and standard deviation for the sampling distribution of the sample proportion of American teenagers with a cell phone are:

• Mean = 0.29
• SD = 0.014

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