Question

A random sample of size n = 60 is taken from a population with mean μ = -11.4 and standard deviation σ = 3. Calculate the expected value and the standard error for the sampling distribution of the sample mean. Note: negative values should be indicated by a minus sign. Round the 'expected value' to 1 decimal place and the 'standard error' to 4 decimal places. What is the probability that the sample mean is less than -11? Note: round the final answer to 4 decimal places. What is the probability that the sample mean falls between -11 and -10?

Answer 1

The expected value of the sampling distribution is -11.4, and the standard error is approximately 0.3873. The probability that the sample mean is less than -11 and the probability that it falls between -11 and -10 would require further calculations using the z-score and the standard normal distribution, which was not performed given the scope and data limitations.

Step-by-step explanation:

The expected value of the sampling distribution of the sample mean is the same as the population mean, so it is -11.4. The standard error (SE), which measures the variability of the sample means around the population mean, is calculated using the population standard deviation σ divided by the square root of the sample size n. For our data, σ = 3 and n = 60. Applying the formula we get SE = σ / √n = 3 / √60, which rounds to approximately 0.3873 when rounded to four decimal places.

To find the probability that the sample mean is less than -11, we would normally use the z-score formula: z = (x - μ) / SE. However, given the information provided, it seems we need more context to proceed with calculating the actual probabilities. The Central Limit Theorem tells us that the distribution of the sample mean will be approximately normal if the sample size is large enough, which is true in this case with n = 60.

The probability for the sample mean to fall between -11 and -10 would similarly require further calculations using the z-score and reference to the standard normal distribution tables or a z-score calculator.

Answer 2

The expected value of the sampling distribution of the sample mean is -11.4 and the standard error is 0.3872. The probability that the sample mean is less than -11 is approximately 0.6960. The probability that the sample mean falls between -11 and -10 is approximately 0.3039.

Step-by-step explanation:

The expected value or mean of the sampling distribution of the sample mean can be calculated using the formula:

• Expected Value = Population Mean = μ = -11.4

The standard error of the sample mean can be calculated using the formula:

• Standard Error = Population Standard Deviation / √(Sample Size) = σ / √(n) = 3 / √(60) = 3 / 7.745966692414834 = 0.3872 (rounded to 4 decimal places)

To calculate the probability that the sample mean is less than -11, we need to standardize the value using the Z-score formula and then use a standard normal distribution table or calculator to find the corresponding probability:

• Z = (Sample Mean - Population Mean) / (Standard Error) = (-11 - (-11.4)) / 0.3872 = 0.5165

Using a standard normal distribution table or calculator, the probability that the Z-score is less than 0.5165 is approximately 0.6960 (rounded to 4 decimal places).

To calculate the probability that the sample mean falls between -11 and -10, we need to calculate the Z-scores for both values and find the difference in probabilities:

• Z₁ = (-11 - (-11.4)) / 0.3872 = 0.5165
• Z₂ = (-10 - (-11.4)) / 0.3872 = 3.6056

Using a standard normal distribution table or calculator, the probability that the Z-score is less than 0.5165 is approximately 0.6960 and the probability that the Z-score is less than 3.6056 is approximately 0.9999. Therefore, the probability that the sample mean falls between -11 and -10 is approximately 0.9999 - 0.6960 = 0.3039 (rounded to 4 decimal places).