Which statement represents the percent of American adults with total cholesterol scores in the borderline-high range?
[tex]$P(-13 \leq z \leq 0)$[/tex]
[tex]$P(-0.77 \leq z \leq 1)$[/tex]
[tex]$P (0 \leq z \leq 1.3)$[/tex]
[tex]$P (0 \leq z \leq 0.77)$[/tex]
Cholesterol levels in the blood are an important health measure. High cholesterol is a risk factor of serious conditions, such as heart disease. This table shows the general guidelines for total cholesterol scores.
| Below [tex]$200 mg / dL$[/tex] | Desirable |
| [tex]$200-239 mg / dL$[/tex] | Borderline high |
| [tex]$240 mg / dL$[/tex] and above | High |
Total cholesterol scores for American adults are normally distributed, with [tex]$\mu=200 mg / dL$[/tex]. Use [tex]$\sigma=$[/tex] [tex]$30 mg / dL$[/tex].
Answer (2)
Calculate the z-score for the lower bound (200 mg/dL): z 1 = 0 .
Calculate the z-score for the upper bound (239 mg/dL): z 2 = 1.3 .
Express the probability of cholesterol scores in the borderline-high range as $P(0
The correct statement representing this probability is: P ( 0 ≤ z ≤ 1.3 ) .
Explanation
Understand the problem and provided data We are given that total cholesterol scores for American adults are normally distributed with a mean ( μ ) of 200 mg/dL and a standard deviation ( σ ) of 30 mg/dL. We want to find the probability that a randomly selected American adult has a total cholesterol score in the borderline-high range, which is defined as 200-239 mg/dL. To do this, we need to convert these cholesterol scores to z-scores.
Calculate the z-score for the lower bound First, let's calculate the z-score for the lower bound of the borderline-high range, which is 200 mg/dL. The z-score formula is: z = σ x − μ where x is the cholesterol score, μ is the mean, and σ is the standard deviation. Plugging in the values, we get: z 1 = 30 200 − 200 = 30 0 = 0 So, the z-score for 200 mg/dL is 0.
Calculate the z-score for the upper bound Next, let's calculate the z-score for the upper bound of the borderline-high range, which is 239 mg/dL. Using the same formula, we get: z 2 = 30 239 − 200 = 30 39 = 1.3 So, the z-score for 239 mg/dL is 1.3.
Express the probability using z-scores Now, we want to find the probability that a randomly selected American adult has a total cholesterol score between 200 mg/dL and 239 mg/dL. This is equivalent to finding the probability that the z-score is between 0 and 1.3. In probability notation, this is written as: P ( 0 ≤ z ≤ 1.3 ) This represents the area under the standard normal curve between z = 0 and z = 1.3.
Select the correct statement Comparing this with the given options, we see that the correct statement is: P ( 0 ≤ z ≤ 1.3 ) This statement represents the percent of American adults with total cholesterol scores in the borderline-high range.
Examples
Understanding cholesterol levels and their distribution is crucial in healthcare. For instance, public health officials can use this information to estimate the number of people at risk for heart disease and to plan appropriate interventions. If we know the percentage of adults with borderline-high cholesterol, we can allocate resources for education and screening programs. For example, if P ( 0 ≤ z ≤ 1.3 ) represents the proportion of adults with borderline-high cholesterol, multiplying this probability by the total adult population gives an estimate of the number of individuals who may benefit from lifestyle changes or medical treatment.
The percent of American adults with total cholesterol scores in the borderline-high range is represented by the statement P ( 0 ≤ z ≤ 1.3 ) , based on the calculated z-scores for 200 mg/dL and 239 mg/dL. This probability notation captures the scores that fall between these two cholesterol levels. Thus, the chosen option for the question is P ( 0 ≤ z ≤ 1.3 ) .
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