You wish to test the following claim (Ha) at a significance level of a = 0.05. H.:µ = 87.3 Ha:µ > 87.3
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
![### Hypothesis Testing of Population Mean
To test a statistical claim about population mean, we are given the following hypothesis:
**Null Hypothesis (H₀):** \( \mu = 87.3 \)
**Alternative Hypothesis (Hₐ):** \( \mu > 87.3 \)
We will conduct this test at a significance level of \( \alpha = 0.05 \).
Given Information:
- Population Standard Deviation (σ): 5.7
- Sample Mean (M): 89
- Sample Size (n): 53
### Step 1: Determine the Critical Value
We need to find the critical value for this test, ensuring it is accurate to three decimal places.
### Step 2: Determine the Test Statistic
We will calculate the test statistic for the given sample. The solution must be accurate to three decimal places.
#### Formulas:
1. **Critical Value**:
- For a one-tailed test with a significance level (\( \alpha \)) of 0.05.
- Using the Z-distribution table:
2. **Test Statistic**:
- The formula for the Z-test in a one-sample test is:
\[
Z = \frac{M - \mu}{\sigma / \sqrt{n}}
\]
Where:
- \( M \) is the sample mean
- \( \mu \) is the population mean as specified in the null hypothesis
- \( \sigma \) is the population standard deviation
- \( n \) is the sample size
### Answers:
**Critical Value:**
**Test Statistic:**
By calculating these values, we proceed with the hypothesis test to determine whether to reject the null hypothesis \( H_0 \) in favor of the alternative hypothesis \( H_a \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3aff424b-0e96-4e90-b1d6-dc40303839e0%2F04dca7b6-5be8-4cdc-aa50-c503e64b37a3%2Fiernulk_processed.jpeg&w=3840&q=75)
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