Question 1200129: The resting heart rate for an adult horse should average about 𝜇 = 43 beats per minute with a (95% of data) range from 14 to 72 beats per minute. Let x be a random variable that represents the resting heart rate for an adult horse. Assume that x has a distribution that is approximately normal.
(a) The empirical rule indicates that for a symmetrical and bell-shaped distribution, approximately 95% of the data lies within two standard deviations of the mean. Therefore, a 95% range of data values extending from 𝜇 - 2𝜎 to 𝜇 + 2𝜎 is often used for "commonly occurring" data values. Note that the interval from 𝜇 - 2𝜎 to 𝜇 + 2𝜎 is 4𝜎 in length. This leads to a "rule of thumb" for estimating the standard deviation from a 95% range of data values:
For a symmetric, bell-shaped distribution:
standard deviation ≈ range/4≈(high value - low value)/4
where it is estimated that about 95% of the commonly occurring data values fall into this range.
Use this "rule of thumb" to estimate the standard deviation of x distribution. (Round your answer to one decimal place.)
beats
(b) What is the probability that the heart rate is less than 25 beats per minute? (Round your answer to four decimal places.)
(c) What is the probability that the heart rate is greater than 60 beats per minute? (Round your answer to four decimal places.)
(d) What is the probability that the heart rate is between 25 and 60 beats per minute? (Round your answer to four decimal places.)
(e) A horse whose resting heart rate is in the upper 13% of the probability distribution of heart rates may have a secondary infection or illness that needs to be treated. What is the heart rate corresponding to the upper 13% cutoff point of the probability distribution? (Round your answer to the nearest whole number.)
Answer by GingerAle(43)   (Show Source):
You can put this solution on YOUR website! **a) Estimate the Standard Deviation**
* **Use the "rule of thumb":**
Standard Deviation ≈ (High Value - Low Value) / 4
Standard Deviation ≈ (72 beats/min - 14 beats/min) / 4
Standard Deviation ≈ 58 beats/min / 4
Standard Deviation ≈ 14.5 beats/min
**b) Probability of Heart Rate Less Than 25 bpm**
* **Standardize the value:**
z = (X - μ) / σ = (25 - 43) / 14.5 ≈ -1.24
* **Find the probability using a z-table:**
P(X < 25) = P(Z < -1.24) ≈ 0.1075
**c) Probability of Heart Rate Greater Than 60 bpm**
* **Standardize the value:**
z = (X - μ) / σ = (60 - 43) / 14.5 ≈ 1.17
* **Find the probability using a z-table:**
P(X > 60) = P(Z > 1.17) ≈ 1 - P(Z < 1.17) ≈ 1 - 0.8790 ≈ 0.1210
**d) Probability of Heart Rate Between 25 and 60 bpm**
* **Find the z-scores:**
* z1 = (25 - 43) / 14.5 ≈ -1.24
* z2 = (60 - 43) / 14.5 ≈ 1.17
* **Find the probability using a z-table:**
P(25 < X < 60) = P(-1.24 < Z < 1.17) = P(Z < 1.17) - P(Z < -1.24) ≈ 0.8790 - 0.1075 ≈ 0.7715
**e) Heart Rate Corresponding to Upper 13%**
* **Find the z-score for the upper 13%:**
* Look up the z-score corresponding to the area to the left of 1 - 0.13 = 0.87 in a z-table.
* z ≈ 1.13
* **Calculate the heart rate:**
* X = μ + (z * σ) = 43 + (1.13 * 14.5) ≈ 61.4
* **Round to the nearest whole number:**
* Heart Rate ≈ 61 beats/min
**Summary:**
* (a) Estimated Standard Deviation: 14.5 beats/min
* (b) Probability of Heart Rate < 25 bpm: 0.1075
* (c) Probability of Heart Rate > 60 bpm: 0.1210
* (d) Probability of Heart Rate between 25 and 60 bpm: 0.7715
* (e) Heart Rate for Upper 13%: 61 beats/min
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