Final answer:
To find the probability that a single randomly selected value is less than 996 dollars, we calculate the z-score and use the standard normal distribution. The probability is approximately 0.5181, or 51.81%. To find the probability that a sample of size n = 98 is randomly selected with a mean less than 996 dollars, we use the Central Limit Theorem. The probability is approximately 0.6736, or 67.36%.
Step-by-step explanation:
To find the probability that a single randomly selected value is less than $996, we can standardize the value and use the standard normal distribution. The formula for standardizing a value is: z = (x - mean) / standard deviation.
Using the given information:
- Mean (μ) = $986
- Standard deviation (σ) = $216
- Value (x) = $996
Substituting the values into the formula: z = (996 - 986) / 216 = 0.0463
Using a standard normal distribution table or a calculator, we can find the probability corresponding to a z-score of 0.0463. The probability is approximately 0.5181, or 51.81%.
Therefore, the probability that a single randomly selected value is less than $996 is 0.5181, or 51.81%.
To find the probability that a sample of size n = 98 is randomly selected with a mean less than $996, we can use the Central Limit Theorem. The Central Limit Theorem states that the distribution of sample means approaches a normal distribution as the sample size increases.
The mean of the sample means (μM) is equal to the population mean (μ) which is $986, and the standard deviation of the sample means (σM) is obtained by dividing the population standard deviation (σ) by the square root of the sample size (√n).
Using the given information:
- Mean (μ) = $986
- Standard deviation (σ) = $216
- Sample size (n) = 98
Substituting the values into the formula: σM = σ / √n = $216 / √98 = $21.8404
Next, we can standardize the value using the formula z = (x - μM) / σM where x is the value of $996.
Substituting the values into the formula: z = (996 - 986) / 21.8404 = 0.4576
We can find the probability corresponding to a z-score of 0.4576 using a standard normal distribution table or a calculator. The probability is approximately 0.6736, or 67.36%.
Therefore, the probability that a sample of size n = 98 is randomly selected with a mean less than $996 is 0.6736, or 67.36%.