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 43​%. Complete parts a through c below.
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Part 1
​a) What would you expect the shape of the sampling distribution for the sample proportion to​ be?
A.
Normal
B.
Uniform
C.
Exponential
D.
None of these

To determine the shape of the sampling distribution of the sample proportion, we use the Central Limit Theorem. According to the theorem, for a large enough sample size, the sampling distribution of the sample proportion is approximately normal if certain conditions are met.

The two main conditions are:
1. The sample size \( n \) is sufficiently large.
2. The expected number of successes (\( np \)) and failures (\( n(1-p) \)) are both greater than 10.

Here:
- \( n = 200 \) (sample size)
- \( p = 0.43 \) (true proportion of smartphone users)
- \( 1 - p = 0.57 \)

We check the conditions:
- \( np = 200 \times 0.43 = 86 \)
- \( n(1-p) = 200 \times 0.57 = 114 \)

Both values (86 and 114) are greater than 10, so the sample size is large enough for the sampling distribution of the sample proportion to be approximately normal.

Thus, the expected shape of the sampling distribution is:

**A. Normal**

Would you like details on how the Central Limit Theorem applies to proportions, or do you have any questions?

Here are 5 related questions to expand on this topic:
1. How would the sampling distribution change if the sample size were smaller?
2. What is the mean and standard deviation of the sampling distribution?
3. How does the population proportion affect the sampling distribution?
4. How can you calculate confidence intervals for the population proportion based on a sample?
5. What role does sample size play in reducing sampling error?

**Tip:** When determining if a sampling distribution is normal, always check if the conditions for the Central Limit Theorem are met, especially \( np \geq 10 \) and \( n(1-p) \geq 10 \).

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 43%. Complete parts a through c below.
​a) What would you expect the shape of the sampling distribution for the sample proportion to​ be?
A. Normal
B. Uniform
C. Exponential
D. None of these

Explore how the Central Limit Theorem applies to estimating the proportion of smart phone users on an investment site. Given a sample size of 200 investors and a true proportion of 43%, the problem examines the expected shape of the sampling distribution. This solution highlights the key conditions for normality, making it ideal for students in Grades 10-12.

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