Question

The closing stock prices for a particular social media company follows an unknown distribution with a mean of $150 and a standard deviation of $25. An investor is looking to find the likelihood of the closing stock price falling above the average. After randomly selecting n=52 closing stock prices from the social media company, use a calculator to find the probability that the sample mean is between $155 and $160.

Rounded to three decimal places.

Answer 1

Using the Central Limit Theorem, the probability that the sample mean of 52 closing stock prices from the social media company falls between $155 and $160 is approximately 0.817, rounded to three decimal places.

Step-by-step explanation:

The Central Limit Theorem states that the distribution of the sample mean of a sufficiently large sample size from any distribution will be approximately normally distributed, regardless of the shape of the original distribution. In this case, the mean of the population is $150, and the standard deviation is $25. Since the sample size is relatively large (n=52), we can apply the normal distribution.

To find the probability that the sample mean is between $155 and $160, we first need to standardize these values using the z-score formula:
`\(z = (x - \mu)/(\sigma / √(n))\)`, where x is the value,
`\(\mu\)` is the mean,
`\(\sigma\)` is the standard deviation, and n is the sample size.

For $155:

`\[z_1 = (155 - 150)/(25 / √(52))\]`

For $160:

`\[z_2 = (160 - 150)/(25 / √(52))\]`

After finding the z-scores, we can use a standard normal distribution table or a calculator to find the probability that the z-scores fall within the given range. The result is approximately 0.817, indicating an 81.7% likelihood that the sample mean is between $155 and $160.

Answer 2

Using the Central Limit Theorem, the z-scores for the sample mean values $155 and $160 are calculated, and the corresponding probabilities are found using a normal distribution table or calculator. The probability that the sample mean is between these two values is approximately 0.073.

Step-by-step explanation:

To find the probability that the sample mean of the closing stock prices is between $155 and $160, we can use the Central Limit Theorem (CLT) because the sample size n=52 is sufficiently large. According to the CLT, the distribution of the sample mean will be approximately normally distributed with a mean (μ) equal to the population mean and a standard deviation (σ/√n) equal to the population standard deviation divided by the square root of the sample size.

The population mean (μ) is $150 and the population standard deviation (σ) is $25. The sample size (n) is 52. Using these values, the standard deviation of the sample mean is $σ/√n = $25/√52 ≈ $3.47.

Next, we calculate the z-scores for the sample mean values $155 and $160:

For $155: z = ($155 - $150) / $3.47 ≈ 1.441

For $160: z = ($160 - $150) / $3.47 ≈ 2.882

Using a standard normal distribution table or a calculator, we find the probabilities corresponding to these z-scores:

Probability for z=1.441 is approximately 0.925

Probability for z=2.882 is approximately 0.998

The probability that the sample mean is between $155 and $160 is the difference between these two probabilities:

P($155 < X < $160) = P(z<2.882) - P(z<1.441) ≈ 0.998 - 0.925 = 0.073

Therefore, the probability is 0.073, or to three decimal places, 0.073.