Questions: Fill in the Blank 5 points The amount of pollutants that are found in waterways near large cities is normally distributed with mean 9.4 ppm and standard deviation 1.6 ppm. 7 randomly selected large cities are studied. Round all answers to 4 decimal places where possible. a. What is the distribution of X ? X ~ N( type your answer... , type your answer... , b. What is the distribution of x̄ ? type your answer... type your answer... c. What is the probability that one randomly selected city's waterway will have more than 8.8 ppm pollutants? d. For the 7 cities, find the probability that the average amount of pollutants is more than 8.8 ppm . type your answer... e. For part d), is the assumption that the distribution is normal necessary? choose your answer... f. Find the IQR for the average of 7 cities. Q1 = type your answer... ppm Q3 = type your answer... ppm IQR: type your answer... ppm

Fill in the Blank 5 points The amount of pollutants that are found in waterways near large cities is normally distributed with mean 9.4 ppm and standard deviation 1.6 ppm. 7 randomly selected large cities are studied. Round all answers to 4 decimal places where possible. a. What is the distribution of X ? X ~ N( type your answer... , type your answer... , b. What is the distribution of x̄ ? type your answer... type your answer... c. What is the probability that one randomly selected city's waterway will have more than 8.8 ppm pollutants? d. For the 7 cities, find the probability that the average amount of pollutants is more than 8.8 ppm . type your answer... e. For part d), is the assumption that the distribution is normal necessary? choose your answer... f. Find the IQR for the average of 7 cities. Q1 = type your answer... ppm Q3 = type your answer... ppm IQR: type your answer... ppm
Transcript text: Fill in the Blank 5 points The amount of pollutants that are found in waterways near large cities is normally distributed with mean 9.4 ppm and standard deviation 1.6 ppm. 7 randomly selected large cities are studied. Round all answers to 4 decimal places where possible. a. What is the distribution of $X$ ? $X \sim N($ type your answer... , $\square$ type your answer... , b. What is the distribution of $\bar{x}$ ? $\square$ type your answer... type your answer... $\square$ c. What is the probability that one randomly selected city's waterway will have more than 8.8 ppm pollutants? $\square$ d. For the 7 cities, find the probability that the average amount of pollutants is more than 8.8 ppm . $\square$ type your answer... e. For part d), is the assumption that the distribution is normal necessary? $\square$ choose your answer... f. Find the IQR for the average of 7 cities. Q1 = type your answer... $\square$ ppm Q3 = type your answer... $\square$ ppm IQR: type your answer... $\square$ ppm
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Solution

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Solution Steps

Step 1: Distribution of \( X \)

The amount of pollutants in waterways near large cities is normally distributed. Therefore, we can express the distribution of \( X \) as: \[ X \sim N(9.4, 2.56) \] where \( 2.56 \) is the variance, calculated as \( \sigma^2 = (1.6)^2 \).

Step 2: Distribution of \( \bar{x} \)

The distribution of the sample mean \( \bar{x} \) for a sample size of \( n = 7 \) is given by: \[ \bar{x} \sim N(9.4, 0.3657) \] where \( 0.3657 \) is the variance of the sample mean, calculated as \( \frac{\sigma^2}{n} = \frac{2.56}{7} \).

Step 3: Z-Score Calculation

To find the probability that one randomly selected city's waterway will have more than \( 8.8 \) ppm pollutants, we first calculate the Z-score: \[ z = \frac{X - \mu}{\sigma} = \frac{8.8 - 9.4}{1.6} = -0.375 \]

Step 4: Probability Calculation

Using the Z-score, we can find the probability: \[ P(X > 8.8) = 1 - P(Z < -0.375) = 1 - 0.6462 = 0.3538 \] Thus, the probability that one city has more than \( 8.8 \) ppm pollutants is \( 0.3538 \).

Final Answer

  • a. The distribution of \( X \) is \( \boxed{N(9.4, 2.56)} \).
  • b. The distribution of \( \bar{x} \) is \( \boxed{N(9.4, 0.3657)} \).
  • c. The probability that one city has more than \( 8.8 \) ppm pollutants is \( \boxed{0.3538} \).
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