4.) Probability of Sampling Proportions

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n

• Distribution Shape

• Sample Size

• Sample Mean

• Population Size

Tags

CCSS.HSS.IC.A.1

CCSS.HSS.IC.B.3

CCSS.HSS.IC.B.4

CCSS.HSS.ID.A.3

CCSS.HSS.ID.A.4

Select the correct description

• Mean of the Sampling Distribution of p-hat

• Population Mean

• Percents of Means

• Proportions

Tags

CCSS.HSS.IC.A.1

CCSS.HSS.IC.B.4

Which pair of formulas is used to find the mean and standard deviation of the sampling distribution of the sample proportion?

•  $\mu_x=n\cdot p$  and  $\sigma_x=\ \sqrt{n\cdot p\cdot\left(1-p\right)}$  

•  $\mu_{\ \overline{x}}\ =\ \mu$  and  $\sigma_{\overline{x}}\ =\ \frac{\sigma}{\sqrt{n}}$  

•  $\mu$   and  $\sigma$  

•  $\mu_{p̂}\ =\ p$   and  $\sigma_{p̂}=\ \sqrt{\frac{p\left(1-p\right)}{n}}$  

Which of the following condition(s) must be met for the sampling distribution of sample proportions to be approximately normal.  

MARK ALL THAT APPLY

•  $n\cdot p\ \ge10$  

•  $n\cdot\left(1-p\right)\ge10$  

• Population is approximately Normal

•  $n\ge30$  

The highlighted region represents...

(Mark all that apply)

• The area to the left of the z-score

• The z-score

• The probability of an outcome that is less than the z-score

• The probability of an outcome that is greater than the z-score

• The area to the right of the z-score

The highlighted region represents...

(Mark all that apply)

• The area to the left of the z-score

• The z-score

• The probability of an outcome that is less than the z-score

• The probability of an outcome that is greater than the z-score

• The area to the right of the z-score

According to the town gardening association, 86% of the over 15,000 families with vegetable gardens in the town grow tomatoes. Suppose we took random samples of 100 families each from this population and calculated the proportion  $p̂$  of sampled families who grow tomatoes for each sample.

Does this satisfy the Normal condition (read carefully)?

• Yes,  $100\left(0.86\right)\ =\ 86\ \ge\ 10$  and  $100\left(0.14\right)\ =\ 14\ \ge\ 10$  

• Yes,  $15,000\left(0.86\right)\ =\ 12900\ \ge\ 10$  and  $15,000\left(0.86\right)\ =\ 2,100\ \ge10$  

• Yes,  $15,000\ \ge\ 10$   and  $100\ge\ 10$  

• Yes, because  $n=100\ \ge30$  , but the CLT

• No, one of the conditions fail

The correct answer was
 $100\left(0.86\right)\ =86\ge10\ and\ \ 100\left(0.14\right)=14\ge10$  
The other solutions either used the population estimate  $N=15,000$ instead of  $n=100$   , or used the CLT ( $n=100\ge30$  ) which only works for questions about means and not proportions.

• K

• K

According to the town gardening association, 86% of the over 15,000 families with vegetable gardens in the town grow tomatoes. Suppose we took random samples of 100 families each from this population and calculated the proportion  $p̂$  of sampled families who grow tomatoes for each sample. (We will soon find the probability that a sample will have 90% or more gardening tomatoes.)

What is the mean of the sampling distribution of  $p̂$  ?

•  $\mu_{p̂}\ =\ 0.86$  

•  $\mu_{\overline{x}}\ =\ 0.90$  

•  $p̂\ =\ 0.86$  

•  $\overline{x}\ =\ 12,900$  

•  $\mu_{p̂}\ =\ 90\%$