To find the probability of the sample mean being between $155 and $160, we first calculate the standard error, convert $155 and $160 to z-scores, and then find the respective probabilities from the standard normal distribution. The required probability is the difference between these two probabilities.
Step-by-step explanation:
We need to find the probability that the mean closing stock price of a social media company is between $155 and $160 when randomly selecting n=52 closing stock prices. We first assume the distribution of the sample mean is approximately normal due to the Central Limit Theorem, which states that the sampling distribution of the sample mean will be normal or nearly normal if the sample size is large enough (n ≥ 30).
Step-by-Step Calculation:
Identify the given values: the population mean (μ) is $150, the population standard deviation (σ) is $25, and the sample size (n) is 52.
Calculate the standard error (SE) of the sample mean: SE = σ / √n, which is $25 / √52.
Convert the target range to z-scores:
For $155: z = ($155 - μ) / SE
For $160: z = ($160 - μ) / SE
Use a standard normal distribution table or a calculator with normal distribution functions to find the probability corresponding to the two z-scores.
The probability that the sample mean is between $155 and $160 is the difference between the two probabilities found in step 4.
The calculations for the z-scores and the probabilities require a calculator as they involve non-trivial numerical computation.