Based on a smartphone​ survey, assume that 52​% of adults with smartphones use them in theaters. In a separate survey of 264 adults with​ smartphones, it is found that 134 use them in theaters.
a. If the 52​% rate is​ correct, find the probability of getting 134 or fewer smartphone owners who use them in theaters.
b. Is the result of 134 significantly​ low?
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Part 1


This problem involves calculating the probability of observing **134 or fewer adults using smartphones in theaters** given that the true population proportion is 52%. We will solve it using **the normal approximation to the binomial distribution**.

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### Part (a): Find the Probability of Getting 134 or Fewer Smartphone Users

#### 1. Define the variables
- Sample size: \( n = 264 \)
- Proportion of smartphone users who use them in theaters: \( p = 0.52 \)
- Complement: \( q = 1 - p = 0.48 \)
- Observed number: \( X = 134 \)

#### 2. Find the mean and standard deviation
Using the formulas for the mean \(\mu\) and standard deviation \(\sigma\) of a binomial distribution:

\[
\mu = n \cdot p = 264 \cdot 0.52 = 137.28
\]

\[
\sigma = \sqrt{n \cdot p \cdot q} = \sqrt{264 \cdot 0.52 \cdot 0.48} = \sqrt{65.8944} \approx 8.12
\]

#### 3. Apply the normal approximation to the binomial distribution
We can approximate the binomial distribution using a normal distribution with mean \( \mu \) and standard deviation \( \sigma \). We also apply the **continuity correction** by subtracting 0.5 from 134 to account for the discrete nature of the binomial distribution.

Thus, we need to find:

\[
P(X \leq 134) = P\left(Z \leq \frac{134 - \mu}{\sigma}\right)
\]

#### 4. Standardize the value using the Z-score formula

\[
Z = \frac{134 - 137.28}{8.12} = \frac{-3.28}{8.12} \approx -0.40
\]

#### 5. Find the probability from the Z-table
Using a Z-table, we look up the probability corresponding to \( Z = -0.40 \). 

\[
P(Z \leq -0.40) \approx 0.3446
\]

So, the probability of getting **134 or fewer smartphone users** who use them in theaters is approximately:

\[
P(X \leq 134) \approx 0.3446
\]

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### Part (b): Is the result of 134 significantly low?

To determine if 134 is significantly low, we compare its Z-score with typical significance levels. Common thresholds are **Z = -1.96** for a 5% significance level (two-tailed) or **Z = -1.645** for a 5% significance level (one-tailed).

- **Z-score** we found: \( Z \approx -0.40 \)

Since \( -0.40 \) is not below these critical Z-scores, **134 is not significantly low**. There is a reasonable chance (34.46%) of getting 134 or fewer smartphone users, assuming the true proportion is 52%.

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### Final Answer
**(a)** The probability of getting 134 or fewer smartphone users who use them in theaters is approximately **0.3446**.  
**(b)** No, the result of 134 is **not significantly low**.

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Would you like more details or have any follow-up questions?  

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Here are 5 related questions you might find interesting:  
1. How does the continuity correction affect normal approximation in binomial distributions?  
2. What is the difference between a one-tailed and two-tailed Z-test?  
3. How would the answer change if the proportion was 60% instead of 52%?  
4. When should we use the normal approximation to the binomial distribution?  
5. How do changes in sample size affect the mean and standard deviation in binomial distributions?  

**Tip:** If the sample size is large and the success probability is close to 0.5, the normal approximation becomes more accurate.

Based on a smartphone​ survey, assume that 52​% of adults with smartphones use them in theaters. In a separate survey of 264 adults with​ smartphones, it is found that 134 use them in theaters.

a. If the 52​% rate is​ correct, find the probability of getting 134 or fewer smartphone owners who use them in theaters.

b. Is the result of 134 significantly​ low?

Calculate the probability of observing 134 or fewer smartphone users in theaters out of 264, assuming a 52% usage rate. Learn how to apply the normal approximation to the binomial distribution, with a step-by-step explanation, and determine if 134 is significantly low. Suitable for college-level statistics students.

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