Answer :
The percentiles for the weight scores 140 lbs, 115 lbs, and 107 lbs are 88.73%, 11.91%, and 28.43% respectively.
We are given that the mean weight among a group of female high school seniors = 120 lbs
Standard deviation (SD) = 16.49 lbs
Now, we need to find percentiles for the following weight scores.
1. Weight Score = 140 lbs (P₁)
2. Weight Score = 115 lbs (P₂)
3. Weight Score = 107 lbs (P₃)
1. Weight Score = 140 lbs (P₁)
Z score can be calculated using the formula:
Z = (X - μ) / σ,
where X is the value of the random variable, μ is the mean, and σ is the standard deviation of the population.
Z₁ = (140 - 120) / 16.49
Z₁ = 1.21
The area under the curve for Z₁ = 1.21 is 0.8873.
P₁ = 0.8873 × 100
P₁ = 88.73%
Therefore, the percentile for 140 lbs is 88.73%.
2. Weight Score = 115 lbs (P₂)
Z₂ = (115 - 120) / 16.49
Z₂ = -0.3032
The area under the curve for Z₂ = -0.3032 is 0.3809.
P₂ = 0.5 - 0.3809
P₂ = 0.1191 × 100
P₂ = 11.91%
Therefore, the percentile for 115 lbs is 11.91%.
3. Weight Score = 107 lbs (P₃)
Z₃ = (107 - 120) / 16.49
Z₃ = -0.7883
The area under the curve for Z₃ = -0.7883 is 0.2157
P₃ = 0.5 - 0.2157
P₃ = 0.2843 × 100
P₃ = 28.43%
Therefore, the percentile for 107 lbs is 28.43%.
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The percentiles for the given weight scores are:
A. 140 lbs.: 88.56%
B. 115 lbs.: 38.21%
C. 107 lbs.: 21.44%
To find the percentiles for the given weight scores, we will use the z-score formula:
z = (x - μ) / σ
where z is the z-score, x is the weight score, μ is the mean, and σ is the standard deviation.
For the given problem, the mean weight (μ) is 120 lbs., and the standard deviation (σ) is 16.49 lbs.
A. For a weight score of 140 lbs.:
1. Calculate the z-score: z = (140 - 120) / 16.49 ≈ 1.21
2. Look up the z-score in a z-table or use a calculator to find the percentile: ≈ 88.56%
B. For a weight score of 115 lbs.:
1. Calculate the z-score: z = (115 - 120) / 16.49 ≈ -0.30
2. Look up the z-score in a z-table or use a calculator to find the percentile: ≈ 38.21%
C. For a weight score of 107 lbs.:
1. Calculate the z-score: z = (107 - 120) / 16.49 ≈ -0.79
2. Look up the z-score in a z-table or use a calculator to find the percentile: ≈ 21.44%
So, the percentiles for the given weight scores are:
A. 140 lbs.: 88.56%
B. 115 lbs.: 38.21%
C. 107 lbs.: 21.44%
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