A random sample of size n = 32 is taken from a population with mean μ = −6.1 and standard deviation σ = 2. [You may find it useful to reference the z table.]
a. Calculate the expected value and the standard error for the sampling distribution of the sample mean. (Negative values should be indicated by a minus sign. Round "expected value" to 1 decimal place and "standard error" to 4 decimal places.)
b. What is the probability that the sample mean is less than −6? (Round final answer to 4 decimal places.)
c. What is the probability that the sample mean falls between −6 and −5? (Round final answer to 4 decimal places.)
a. The expected value of the sampling distribution of the sample mean is equal to the population mean, which in this case is μ = -6.1.
The standard error of the sampling distribution of the sample mean is calculated using the formula: σ/√n, where σ is the population standard deviation and n is the sample size.
In this case, the population standard deviation is σ = 2 and the sample size is n = 32. Plugging these values into the formula, we get:
Standard error = 2/√32 = 0.3542
Therefore, the expected value of the sampling distribution of the sample mean is -6.1 and the standard error is 0.3542.
b. To calculate the probability that the sample mean is less than -6, we need to standardize the value of -6 using z-score.
The z-score formula is: z = (x - μ) / (σ/√n), where x is the value we want to calculate the probability for.
In this case, x = -6, μ = -6.1, σ = 2, and n = 32. Plugging these values into the formula, we get:
z = (-6 - (-6.1)) / (2/√32) = -0.1 / 0.3542 = -0.2823
Looking up the z-score in the z-table, we find that the corresponding probability is 0.3888.
Therefore, the probability that the sample mean is less than -6 is 0.3888.
c. To calculate the probability that the sample mean falls between -6 and -5, we need to calculate the z-scores for both values and find the difference between their corresponding probabilities.
For -6:
z = (-6 - (-6.1)) / (2/√32) = -0.1 / 0.3542 = -0.2823
For -5:
z = (-5 - (-6.1)) / (2/√32) = 1.1 / 0.3542 = 3.107
Looking up both z-scores in the z-table, we find that the corresponding probabilities are:
For -6: 0.3888
For -5: 0.9994
The probability that the sample mean falls between -6 and -5 is the difference between these two probabilities:
0.9994 - 0.3888 = 0.6106
Therefore, the probability that the sample mean falls between -6 and -5 is 0.6106.
To calculate the expected value and standard error of the sampling distribution of the sample mean, we can use the formulas:
Expected Value = μ = -6.1
Standard Error = σ / √n = 2 / √32 = 2 / 5.6569 ≈ 0.3536
a. The expected value for the sampling distribution of the sample mean is -6.1. The standard error is approximately 0.3536.
To calculate the probability that the sample mean is less than -6, we need to calculate the z-score and use the z-table.
b. First, calculate the z-score:
z = (x - μ) / (σ / √n) = (-6 - (-6.1)) / (2 / √32) ≈ 0.1414
Next, look up the z-score in the z-table to find the corresponding area/probability:
Area = 0.5557
The probability that the sample mean is less than -6 is approximately 0.5557.
c. To calculate the probability that the sample mean falls between -6 and -5, we need to calculate the z-scores for each boundary and use the z-table.
For -6:
z1 = (-6 - (-6.1)) / (2 / √32) ≈ 0.1414
For -5:
z2 = (-5 - (-6.1)) / (2 / √32) ≈ 1.0707
Next, look up the z-scores in the z-table to find the corresponding areas for each boundary:
Area1 = 0.5557
Area2 = 0.8577
To find the probability that the sample mean falls between -6 and -5, subtract the area for the lower boundary from the area for the upper boundary:
Probability = Area2 - Area1 = 0.8577 - 0.5557 ≈ 0.302
The probability that the sample mean falls between -6 and -5 is approximately 0.302.
To calculate the expected value and standard error for the sampling distribution of the sample mean, we can use the formulas:
Expected Value (μ): The expected value of the sample mean is equal to the population mean, so in this case, the expected value is -6.1.
Standard Error (SE): The standard error of the sample mean is equal to the population standard deviation divided by the square root of the sample size. Therefore, we can use the formula:
SE = σ / √n
where σ is the population standard deviation (2) and n is the sample size (32).
Applying the formula:
SE = 2 / √32 ≈ 0.3536 (rounded to 4 decimal places)
Therefore, the expected value of the sample mean is -6.1, and the standard error is approximately 0.3536.
b. To find the probability that the sample mean is less than -6, we can use the Z-score and the Z-table. The Z-score is calculated using the formula:
Z = (X - μ) / (σ / √n)
where X is the value we want to find the probability for, μ is the population mean, σ is the population standard deviation, and n is the sample size.
In this case, we want to find the probability for X = -6, μ = -6.1, σ = 2, and n = 32.
Calculating the Z-score:
Z = (-6 - (-6.1)) / (2 / √32) ≈ -1.8974
Using the Z-table, we can find that the area to the left of Z = -1.8974 is approximately 0.0286.
Therefore, the probability that the sample mean is less than -6 is approximately 0.0286 (rounded to 4 decimal places).
c. To find the probability that the sample mean falls between -6 and -5, we need to calculate the Z-scores for both -6 and -5, and then find the difference between their respective areas under the curve.
Calculating the Z-score for -6:
Z1 = (-6 - (-6.1)) / (2 / √32) ≈ -1.8974
Calculating the Z-score for -5:
Z2 = (-5 - (-6.1)) / (2 / √32) ≈ -0.5987
Using the Z-table, we can find the area to the left of Z = -1.8974 is approximately 0.0286, and the area to the left of Z = -0.5987 is approximately 0.2757.
To find the probability between -6 and -5, we subtract the area for Z = -0.5987 from the area for Z = -1.8974:
0.0286 - 0.2757 = -0.2471
Therefore, the probability that the sample mean falls between -6 and -5 is approximately 0.2471 (rounded to 4 decimal places).