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Answer :
Let's solve the problem step by step, considering the given scenario:
a. Finding the Probability:
1. Identify the Parameters:
- We know that [tex]\( 51\% \)[/tex] of adults with smartphones use them in theaters, so the population proportion ([tex]\( p \)[/tex]) is 0.51.
- The sample size ([tex]\( n \)[/tex]) is 250.
- We observe that 109 adults in the sample use smartphones in theaters.
2. Calculate the Mean and Standard Deviation:
- The mean ([tex]\( \mu \)[/tex]) of the sampling distribution for the number of adults using smartphones in theaters is:
[tex]\[
\mu = n \times p = 250 \times 0.51 = 127.5
\][/tex]
- The standard deviation ([tex]\( \sigma \)[/tex]) of this distribution is calculated by:
[tex]\[
\sigma = \sqrt{n \times p \times (1-p)} = \sqrt{250 \times 0.51 \times (1-0.51)} \approx 7.904
\][/tex]
3. Calculate the Z-score:
- The Z-score helps us understand how far the observed value (109) is from the expected mean in terms of standard deviations. The formula is:
[tex]\[
Z = \frac{x - \mu}{\sigma} = \frac{109 - 127.5}{7.904} \approx -2.34
\][/tex]
4. Find the Probability:
- Use the standard normal distribution to find the probability of getting 109 or fewer smartphone users. The cumulative probability corresponding to a Z-score of [tex]\(-2.34\)[/tex] is about 0.00963. This means there's a 0.963% chance of observing 109 or fewer users if the true proportion is 51%.
b. Determine if the Result is Significantly Low:
- A result is typically considered significantly low if the probability is less than 5% (0.05).
- In this case, the probability of 0.00963 is indeed less than 0.05, suggesting that getting 109 or fewer users is significantly low.
Thus, based on the analysis, the probability of observing 109 or fewer adults using smartphones in theaters, when 51% is the true rate, is approximately 0.963%, and this result is statistically significantly low.
a. Finding the Probability:
1. Identify the Parameters:
- We know that [tex]\( 51\% \)[/tex] of adults with smartphones use them in theaters, so the population proportion ([tex]\( p \)[/tex]) is 0.51.
- The sample size ([tex]\( n \)[/tex]) is 250.
- We observe that 109 adults in the sample use smartphones in theaters.
2. Calculate the Mean and Standard Deviation:
- The mean ([tex]\( \mu \)[/tex]) of the sampling distribution for the number of adults using smartphones in theaters is:
[tex]\[
\mu = n \times p = 250 \times 0.51 = 127.5
\][/tex]
- The standard deviation ([tex]\( \sigma \)[/tex]) of this distribution is calculated by:
[tex]\[
\sigma = \sqrt{n \times p \times (1-p)} = \sqrt{250 \times 0.51 \times (1-0.51)} \approx 7.904
\][/tex]
3. Calculate the Z-score:
- The Z-score helps us understand how far the observed value (109) is from the expected mean in terms of standard deviations. The formula is:
[tex]\[
Z = \frac{x - \mu}{\sigma} = \frac{109 - 127.5}{7.904} \approx -2.34
\][/tex]
4. Find the Probability:
- Use the standard normal distribution to find the probability of getting 109 or fewer smartphone users. The cumulative probability corresponding to a Z-score of [tex]\(-2.34\)[/tex] is about 0.00963. This means there's a 0.963% chance of observing 109 or fewer users if the true proportion is 51%.
b. Determine if the Result is Significantly Low:
- A result is typically considered significantly low if the probability is less than 5% (0.05).
- In this case, the probability of 0.00963 is indeed less than 0.05, suggesting that getting 109 or fewer users is significantly low.
Thus, based on the analysis, the probability of observing 109 or fewer adults using smartphones in theaters, when 51% is the true rate, is approximately 0.963%, and this result is statistically significantly low.
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