Suppose you have a normal population of quiz scores, with mean 40 and standard deviation 10. Select a random sample of 40. What is the chance that the mean of the quiz scores will not exceed 45? Use a sampling distribution of sample means to calculate the standard error; fill in the blank below.
Transcribed Image Text:The expression shown is a standard normal probability calculation:
\[
P(\overline{X} < 45) = P \left( Z < \frac{45 - (\ )}{10 / \sqrt{(\ )}} \right) = P(z < 3.16) = (\ )
\]
This equation involves calculating the probability that a sample mean \(\overline{X}\) is less than 45 for a normally distributed variable, converted into a standard normal variable \(Z\).
- \( P(\overline{X} < 45) \) represents the probability that the sample mean is less than 45.
- The calculation transforms this into a standard normal probability using the Z-score formula.
- The Z-score formula \[\frac{45 - (\ )}{10 / \sqrt{(\ )}}\] simplifies to \[z < 3.16\].
- The outcome would typically reference a Z-table or a statistical software output for the probability \(P(z < 3.16)\).
This process is commonly used in statistics for hypothesis testing or confidence interval estimation.
Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...
![The expression shown is a standard normal probability calculation:
\[
P(\overline{X} < 45) = P \left( Z < \frac{45 - (\ )}{10 / \sqrt{(\ )}} \right) = P(z < 3.16) = (\ )
\]
This equation involves calculating the probability that a sample mean \(\overline{X}\) is less than 45 for a normally distributed variable, converted into a standard normal variable \(Z\).
- \( P(\overline{X} < 45) \) represents the probability that the sample mean is less than 45.
- The calculation transforms this into a standard normal probability using the Z-score formula.
- The Z-score formula \[\frac{45 - (\ )}{10 / \sqrt{(\ )}}\] simplifies to \[z < 3.16\].
- The outcome would typically reference a Z-table or a statistical software output for the probability \(P(z < 3.16)\).
This process is commonly used in statistics for hypothesis testing or confidence interval estimation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F833b74ec-a747-4e93-bb2e-129456930c98%2F1f8912fc-5b70-4512-96c9-d820b7c545ab%2F88661f_processed.jpeg&w=3840&q=75)
