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.
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.)
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