An investment website can tell what devices are used to access the site. The site managers wonder whether they should enhance the facilities for trading via​ "smart phones", so they want to estimate the proportion of users who access the site that way​ (even if they also use their computers​ sometimes). They draw a random sample of

200200

investors from their customers. Suppose that the true proportion of smart phone users is

3131​%.

Question content area bottom

Part 1

​a) What would the standard deviation of the sampling distribution of the proportion of the smart phone users​ be?

0.0330.033

​(Round to three decimal places as​ needed.)

Part 2

​b) What is the probability that the sample proportion of smart phone users is greater than

0.310.31​?

0.5000.500

​(Round to three decimal places as​ needed.)

Part 3

​c) What is the probability that the sample proportion is between

0.270.27

and

0.350.35​?

0.7740.774

​(Round to three decimal places as​ needed.)

Part 4

​d) What is the probability that the sample proportion is less than

0.250.25​?

enter your response here

​(Round to three decimal places as​ needed.)

### Part 1:
**a) What would the standard deviation of the sampling distribution of the proportion of the smart phone users be?**

The standard deviation of the sampling distribution of a proportion is given by the formula:

\[
\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}
\]

Where:
- \( p \) = true proportion of the population (given as 0.31),
- \( n \) = sample size (given as 200).

\[
\sigma_{\hat{p}} = \sqrt{\frac{0.31(1 - 0.31)}{200}} = \sqrt{\frac{0.31 \times 0.69}{200}} = \sqrt{\frac{0.2139}{200}} = \sqrt{0.0010695} \approx 0.033
\]

So, the standard deviation is **0.033**.

---

### Part 2:
**b) What is the probability that the sample proportion of smart phone users is greater than 0.31?**

For this, we need to find the Z-score and use the standard normal distribution table. The Z-score is calculated as:

\[
Z = \frac{\hat{p} - p}{\sigma_{\hat{p}}}
\]

Where:
- \( \hat{p} = 0.31 \) (the sample proportion),
- \( p = 0.31 \) (the true proportion),
- \( \sigma_{\hat{p}} = 0.033 \).

\[
Z = \frac{0.31 - 0.31}{0.033} = 0
\]

The Z-score is 0, and using the standard normal distribution table, the probability \( P(Z > 0) \) is **0.500** (since the distribution is symmetric).

---

### Part 3:
**c) What is the probability that the sample proportion is between 0.27 and 0.35?**

First, we calculate the Z-scores for both 0.27 and 0.35.

For \( \hat{p} = 0.27 \):

\[
Z = \frac{0.27 - 0.31}{0.033} = \frac{-0.04}{0.033} \approx -1.212
\]

For \( \hat{p} = 0.35 \):

\[
Z = \frac{0.35 - 0.31}{0.033} = \frac{0.04}{0.033} \approx 1.212
\]

Now, using the Z-scores in the standard normal distribution table:

- \( P(Z < -1.212) \approx 0.113 \),
- \( P(Z < 1.212) \approx 0.887 \).

The probability between 0.27 and 0.35 is:

\[
P(0.27 < \hat{p} < 0.35) = P(Z < 1.212) - P(Z < -1.212) = 0.887 - 0.113 = 0.774
\]

So, the probability is **0.774**.

---

### Part 4:
**d) What is the probability that the sample proportion is less than 0.25?**

We calculate the Z-score for \( \hat{p} = 0.25 \):

\[
Z = \frac{0.25 - 0.31}{0.033} = \frac{-0.06}{0.033} \approx -1.818
\]

Using the standard normal distribution table:

- \( P(Z < -1.818) \approx 0.035 \).

So, the probability is **0.035**.

---

### Summary of answers:
- **a)** Standard deviation: \( 0.033 \)
- **b)** Probability \( P(\hat{p} > 0.31) = 0.500 \)
- **c)** Probability \( P(0.27 < \hat{p} < 0.35) = 0.774 \)
- **d)** Probability \( P(\hat{p} < 0.25) = 0.035 \)

Would you like further details or clarification on any part?

---

Here are 5 related questions:
1. How would the standard deviation change if the sample size were doubled?
2. What would happen to the probability in part (b) if the true proportion were higher?
3. How is the Z-score used in real-world statistics for decision-making?
4. What does the probability in part (d) imply about the sample's distribution?
5. How would increasing the sample size affect the confidence interval for the proportion?

**Tip:** The standard deviation of a sample proportion decreases as the sample size increases, which leads to more precise estimates.

An investment website can tell what devices are used to access the site. The site managers wonder whether they should enhance the facilities for trading via 'smart phones', so they want to estimate the proportion of users who access the site that way (even if they also use their computers sometimes). They draw a random sample of 200 investors from their customers. Suppose that the true proportion of smart phone users is 31%. 

Part 1: a) What would the standard deviation of the sampling distribution of the proportion of the smart phone users be?
0.033 (Round to three decimal places as needed.)

Part 2: b) What is the probability that the sample proportion of smart phone users is greater than 0.31?
0.500 (Round to three decimal places as needed.)

Part 3: c) What is the probability that the sample proportion is between 0.27 and 0.35?
0.774 (Round to three decimal places as needed.)

Part 4: d) What is the probability that the sample proportion is less than 0.25?

Calculate the standard deviation of the sampling distribution of smart phone users and find probabilities for different scenarios using the Z-score. This statistics problem involves estimating proportions and applying the Central Limit Theorem. Suitable for college-level statistics students.

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