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Answer :
Let's solve the problem step-by-step:
a. Calculate the probability of getting 109 or fewer smartphone owners who use them in theaters, given the 51% rate.
1. Proportion of adults using smartphones in theaters: According to the LG smartphone survey, 51% of adults use their smartphones in theaters. This means the probability [tex]\( p \)[/tex] is 0.51.
2. Sample size: There are 250 adults surveyed, so the sample size [tex]\( n \)[/tex] is 250.
3. Sample proportion: In a separate survey, 109 out of 250 adults use their smartphones in theaters. The sample proportion [tex]\( \hat{p} \)[/tex] is calculated as:
[tex]\[
\hat{p} = \frac{109}{250} = 0.436
\][/tex]
4. Standard deviation of the sample proportion: The standard deviation for the sample proportion is calculated using the formula for the standard deviation of a proportion:
[tex]\[
\sigma = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.51 \times (1 - 0.51)}{250}} \approx 0.0316
\][/tex]
5. Calculate the z-score: The z-score tells us how many standard deviations our sample proportion is from the population proportion. The formula for the z-score is:
[tex]\[
z = \frac{\hat{p} - p}{\sigma} = \frac{0.436 - 0.51}{0.0316} \approx -2.34
\][/tex]
6. Calculate the probability: Use the z-score to find the probability of obtaining a sample proportion of 109 or fewer using the standard normal distribution. The probability corresponds to the cumulative distribution function (CDF) value for the z-score:
[tex]\[
\text{Probability} = P(Z \leq -2.34) \approx 0.0096
\][/tex]
b. Is the result of 109 significantly low?
To determine if the result of 109 adults using smartphones in theaters is significantly low, we compare the probability from part (a) to a common significance level. A typical threshold for significance is 0.05.
- Probability found: [tex]\( \approx 0.0096 \)[/tex]
Since the probability [tex]\( 0.0096 \)[/tex] is less than 0.05, the result of 109 is considered significantly low.
Comment on the LG smartphone survey:
The result of 109 smartphone users in theaters from the second survey is significantly lower than expected if the true proportion were 51%. This might suggest that the original survey's 51% rate may not accurately represent the behavior of the broader population, or there could have been differences in the sample or method that led to this discrepancy.
a. Calculate the probability of getting 109 or fewer smartphone owners who use them in theaters, given the 51% rate.
1. Proportion of adults using smartphones in theaters: According to the LG smartphone survey, 51% of adults use their smartphones in theaters. This means the probability [tex]\( p \)[/tex] is 0.51.
2. Sample size: There are 250 adults surveyed, so the sample size [tex]\( n \)[/tex] is 250.
3. Sample proportion: In a separate survey, 109 out of 250 adults use their smartphones in theaters. The sample proportion [tex]\( \hat{p} \)[/tex] is calculated as:
[tex]\[
\hat{p} = \frac{109}{250} = 0.436
\][/tex]
4. Standard deviation of the sample proportion: The standard deviation for the sample proportion is calculated using the formula for the standard deviation of a proportion:
[tex]\[
\sigma = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.51 \times (1 - 0.51)}{250}} \approx 0.0316
\][/tex]
5. Calculate the z-score: The z-score tells us how many standard deviations our sample proportion is from the population proportion. The formula for the z-score is:
[tex]\[
z = \frac{\hat{p} - p}{\sigma} = \frac{0.436 - 0.51}{0.0316} \approx -2.34
\][/tex]
6. Calculate the probability: Use the z-score to find the probability of obtaining a sample proportion of 109 or fewer using the standard normal distribution. The probability corresponds to the cumulative distribution function (CDF) value for the z-score:
[tex]\[
\text{Probability} = P(Z \leq -2.34) \approx 0.0096
\][/tex]
b. Is the result of 109 significantly low?
To determine if the result of 109 adults using smartphones in theaters is significantly low, we compare the probability from part (a) to a common significance level. A typical threshold for significance is 0.05.
- Probability found: [tex]\( \approx 0.0096 \)[/tex]
Since the probability [tex]\( 0.0096 \)[/tex] is less than 0.05, the result of 109 is considered significantly low.
Comment on the LG smartphone survey:
The result of 109 smartphone users in theaters from the second survey is significantly lower than expected if the true proportion were 51%. This might suggest that the original survey's 51% rate may not accurately represent the behavior of the broader population, or there could have been differences in the sample or method that led to this discrepancy.
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