A recent study of US children under 11 years old found that they spend on average 142 minutes a day watching tv. Assuming normal distribution with the standard deviation of 44 minutes, answer the following.
A) what percent of children of this age spends over 90 minutes per day watching tv?
B) find and interpret P70, the 70th - percentile.
To answer these questions, we will use the concept of standard deviation and the properties of a normal distribution. Here are the steps to solve each part:
A) Finding the percentage of children who spend over 90 minutes per day watching TV:
1. Calculate the z-score: The z-score tells us how many standard deviations a value is from the mean. We can calculate the z-score using the formula:
z = (x - μ) / σ
Where:
- x is the value we want to convert to a z-score (90 minutes in this case),
- μ is the mean (142 minutes),
- σ is the standard deviation (44 minutes).
Plugging in the values: z = (90 - 142) / 44 = -1.18
2. Find the area under the normal curve to the right of the z-score: The z-score represents the number of standard deviations below or above the mean. In this case, since we are interested in values over 90 minutes, we need to find the area under the curve to the right of the z-score. We can refer to the z-table or use a statistical calculator to find this area. In this case, the area to the left of the z-score is approximately 0.1190 (using the z-table).
3. Calculate the percentage: Subtract the area found in step 2 from 1 (or multiply by 100) to find the percentage of children who spend over 90 minutes per day watching TV.
Percentage = (1 - 0.1190) × 100 ≈ 88.10%
Therefore, approximately 88.10% of children of this age spend over 90 minutes per day watching TV.
B) Finding and interpreting P70, the 70th percentile:
The 70th percentile (P70) represents the value below which 70% of the data falls. To find the corresponding value for P70, we can use the z-score formula:
1. Convert 70% to a z-score: The z-score formula is given by:
z = (x - μ) / σ
We need to find the z-score for the given percentile, so we set z = z-score and solve for x.
Rearranging the formula: x = z * σ + μ
2. Calculate the value of x using the formula: P70 = z * σ + μ
Given that P70 represents the 70th percentile, we can use the z-table or a statistical calculator to find the z-score associated with it. Let's say the z-score is -0.53 for P70 (using the z-table).
Plugging in the values: x = -0.53 * 44 + 142 ≈ 118.68
Therefore, P70 (the 70th percentile) is approximately 118.68 minutes.
Interpretation: P70 indicates that 70% of children spend 118.68 minutes or less watching TV per day.