What is the probability that a sophomore prefers a smaller smartphone? What is the probability that a freshman prefers a smaller smartphone? Is the probability that a person is a f... What is the probability that a sophomore prefers a smaller smartphone? What is the probability that a freshman prefers a smaller smartphone? Is the probability that a person is a freshman and prefers a larger smartphone the same as the probability that a freshman prefers a larger smartphone? Explain.
Understand the Problem
The question is asking for the probability calculations based on survey results from a group of underclassmen who prefer different sizes of smartphones. It involves statistical concepts and the use of a frequency table to derive probabilities for sophomores and freshmen regarding their preferences for smartphone sizes.
Answer
The probability that a sophomore prefers a smaller smartphone is \( \frac{36}{98} \), the probability that a freshman prefers a smaller smartphone is \( \frac{32}{92} \), and the probability that a freshman prefers a larger smartphone is \( \frac{60}{92} \).
Answer for screen readers
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The probability that a sophomore prefers a smaller smartphone is ( P(\text{sophomore prefers smaller}) = \frac{36}{98} \approx 0.367 ).
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The probability that a freshman prefers a smaller smartphone is ( P(\text{freshman prefers smaller}) = \frac{32}{92} \approx 0.348 ).
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The probability that a freshman prefers a larger smartphone is ( P(\text{freshman prefers larger}) = \frac{60}{92} \approx 0.652 ).
Steps to Solve
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Find the total number of sophomores Look at the frequency table. The total number of sophomores is given as 98.
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Calculate the number of sophomores who prefer smaller smartphones According to the table, 36 sophomores prefer smaller smartphones.
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Calculate the probability for sophomores The probability ( P ) that a sophomore prefers a smaller smartphone can be calculated using the formula: $$ P(\text{sophomore prefers smaller}) = \frac{\text{number of sophomores preferring smaller}}{\text{total number of sophomores}} $$ Substituting the values: $$ P(\text{sophomore prefers smaller}) = \frac{36}{98} $$
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Find the total number of freshmen From the table, the total number of freshmen is also 92.
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Calculate the number of freshmen who prefer smaller smartphones According to the table, 32 freshmen prefer smaller smartphones.
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Calculate the probability for freshmen Using the same formula as before: $$ P(\text{freshman prefers smaller}) = \frac{\text{number of freshmen preferring smaller}}{\text{total number of freshmen}} $$ Substituting the values: $$ P(\text{freshman prefers smaller}) = \frac{32}{92} $$
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Determine the probabilities for larger smartphones To find the third probability, we observe that the probability a freshman prefers a larger smartphone is the complement of preferring a smaller smartphone. Therefore, $ P(\text{freshman prefers larger}) = 1 - P(\text{freshman prefers smaller}) $.
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Calculate and compare probabilities Since we already computed $ P(\text{freshman prefers smaller}) $, we can calculate: $$ P(\text{freshman prefers larger}) = 1 - \frac{32}{92} $$
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Explain the relationship Finally, explain whether the probability of being a freshman who prefers a larger smartphone is equal to the probability that a freshman prefers a larger smartphone.
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The probability that a sophomore prefers a smaller smartphone is ( P(\text{sophomore prefers smaller}) = \frac{36}{98} \approx 0.367 ).
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The probability that a freshman prefers a smaller smartphone is ( P(\text{freshman prefers smaller}) = \frac{32}{92} \approx 0.348 ).
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The probability that a freshman prefers a larger smartphone is ( P(\text{freshman prefers larger}) = \frac{60}{92} \approx 0.652 ).
More Information
The calculations show the preferences among freshmen and sophomores about smartphone sizes. These probabilities can help marketers understand the preferences of these underclassmen.
Tips
- Confusing the total number of students with the number preferring a specific smartphone size. Make sure to use the appropriate total for each calculation.
- Miscalculating probabilities by forgetting the total when determining proportions.
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