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Normal Distribution TV Ownership Problem: Find Percentage of People Keeping TV Between 3 and 4 Years
Statistics
High School Grade 11/12
Question Content
A survey found that people keep their television sets an average of 4.8 years, with a standard deviation of 0.89 year. Assuming this follows a normal distribution, what percent of people keep a TV between 3 and 4 years?
Correct Answer
Approximately 15.27% of people keep their TV between 3 and 4 years.
Detailed Solution Steps
1
Step 1: Define $X$ as the number of years a TV is kept. We need to find $P(3 < X < 4)$ where $\\mu = 4.8$ and $\\sigma = 0.89$.
2
Step 2: Calculate z-scores for $X=3$ and $X=4$ using $z = \\frac{X - \\mu}{\\sigma}$. For $X=3$: $z_1 = \\frac{3 - 4.8}{0.89} \\approx -2.02$. For $X=4$: $z_2 = \\frac{4 - 4.8}{0.89} \\approx -0.90$.
3
Step 3: Find the area to the left of $z=-0.90$ (≈0.1841) and the area to the left of $z=-2.02$ (≈0.0217) using a standard normal table or calculator.
4
Step 4: Subtract the smaller area from the larger area to find the area between the two z-scores: $0.1841 - 0.0217 = 0.1524$, or approximately 15.27% (rounded to two decimal places).
Knowledge Points Involved
1
Z-Score Calculation for Interval Data
When finding the probability of a raw score falling between two values in a normal distribution, convert both raw scores to z-scores. The area between the two z-scores is found by subtracting the cumulative area of the lower z-score from the cumulative area of the higher z-score.
2
Cumulative Area in Standard Normal Distribution
The cumulative area to the left of a z-score on the standard normal curve represents the percentage of data that is less than or equal to that z-score. It is used to calculate probabilities for ranges of values by subtracting cumulative areas.
3
Application of Normal Distribution to Real-World Data
Normal distributions are often used to model real-world continuous data (like the length of time people keep a TV) that cluster around a mean. By converting raw data to z-scores, we can use standard normal distribution tables to find probabilities for real-world scenarios.
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