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Math  /  Data & Statistics

QuestionAssume the average amount of caffeine consumed daily by adults is normally distributed with a mean of 240 mg and a standard deviation of 46 mg . In a random sample of 300 adults, how many consume at least 320 mg of caffeine daily?
Click here to view page 1 of the standard normal distribution table. Click here to view page 2 of the standard normal distribution table.
Of the 300 adults, approximately □\square adults consume at least 320 mg of caffeine daily. (Round to the nearest whole number as needed.)

Studdy Solution

STEP 1

1. The daily caffeine consumption is normally distributed with a mean (μ\mu) of 240 mg and a standard deviation (σ\sigma) of 46 mg.
2. We are interested in finding the number of adults who consume at least 320 mg of caffeine daily.
3. The sample size is 300 adults.
4. Use the standard normal distribution table to find probabilities.

STEP 2

1. Calculate the z-score for 320 mg.
2. Use the z-score to find the probability of consuming at least 320 mg.
3. Calculate the number of adults consuming at least 320 mg.

STEP 3

Calculate the z-score using the formula:
z=X−μσ z = \frac{X - \mu}{\sigma}
where X=320 X = 320 , μ=240 \mu = 240 , and σ=46 \sigma = 46 .
z=320−24046 z = \frac{320 - 240}{46}
z=8046 z = \frac{80}{46}
z≈1.7391 z \approx 1.7391

STEP 4

Use the standard normal distribution table to find the probability that Z<1.7391 Z < 1.7391 .
The table gives P(Z<1.7391)≈0.9582 P(Z < 1.7391) \approx 0.9582 .

STEP 5

The probability of consuming at least 320 mg is:
P(Z≥1.7391)=1−P(Z<1.7391) P(Z \geq 1.7391) = 1 - P(Z < 1.7391)
P(Z≥1.7391)=1−0.9582 P(Z \geq 1.7391) = 1 - 0.9582
P(Z≥1.7391)≈0.0418 P(Z \geq 1.7391) \approx 0.0418

STEP 6

Calculate the number of adults consuming at least 320 mg:
Number of adults=0.0418×300 \text{Number of adults} = 0.0418 \times 300
Number of adults≈12.54 \text{Number of adults} \approx 12.54
Round to the nearest whole number:
13 \boxed{13}
Of the 300 adults, approximately 13\boxed{13} adults consume at least 320 mg of caffeine daily.

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