Final answer:
a. The probability that the sodium content will be more than 670 mg is 43.75%. b. The probability that the mean of the sample will be larger than 670 mg is 30.94%. c. The probability for part a is greater than that for part b because when we take a sample, the standard deviation becomes smaller due to the effect of the sample size in the formula.
Step-by-step explanation:
a. To find the probability that the sodium content will be more than 670 mg, we need to find the z-score corresponding to 670 mg. The formula to calculate the z-score is: z = (x - μ) / σ Where x is the value, μ is the mean, and σ is the standard deviation. Plugging in the numbers, we get: z = (670 - 665) / 32 = 0.15625. Using a standard normal distribution table or a calculator, we find that the area to the right of a z-score of 0.15625 is approximately 0.4375. Therefore, the probability that the sodium content will be more than 670 mg is 0.4375 or 43.75%.
b. To find the probability that the mean of the sample will be larger than 670 mg, we need to find the z-score corresponding to 670 mg. The formula to calculate the z-score is: z = (x - μ) / (σ/√n) Where x is the value, μ is the mean, σ is the standard deviation, and n is the sample size. Plugging in the numbers, we get: z = (670 - 665) / (32 / √10) = 0.49587. Using a standard normal distribution table or a calculator, we find that the area to the right of a z-score of 0.49587 is approximately 0.3094. Therefore, the probability that the mean of the sample will be larger than 670 mg is 0.3094 or 30.94%.
c. The probability for part a is greater than that for part b because when we take a sample, the standard deviation becomes smaller due to the effect of the sample size in the formula. This results in a narrower distribution, meaning that the probabilities of extreme values decrease. Therefore, the probability of obtaining a sample mean larger than a certain value is smaller compared to the probability of obtaining an individual value larger than the same value.