Question

Conduct a goodness of fit test at a 1% level of significance to see whether the following sample appears to have been selected from a normal distribution. The sample mean and sample standard deviation are 69 and 17.8237 respectively. / Doen ‘n passingsgehalte toets met ‘n 1% betekenispeil om te bepaal of die volgende steekproef gekies is uit ‘n normaalverdeling. Die steekproefgemiddeld en steekproef standaardafwyking is 69 en 17.8237 respektiewelik:
50 80 90 50 55 90 55 50 70 95
90 60 85 58 60 98 62 80 62 40

Answer 1

To conduct a goodness of fit test to determine whether the provided sample appears to be from a normal distribution, we can use the following steps:

### Step 1: Formulate the Hypotheses
- **Null Hypothesis (H₀):** The sample is drawn from a normal distribution.
- **Alternative Hypothesis (H₁):** The sample is not drawn from a normal distribution.

### Step 2: Collect and Organize the Data
The given sample data:
```
50, 80, 90, 50, 55, 90, 55, 50, 70, 95,
90, 60, 85, 58, 60, 98, 62, 80, 62, 40
```

### Step 3: Determine Sample Statistics
From the problem:
- Sample Mean (M) = 69
- Sample Standard Deviation (s) = 17.8237

### Step 4: Determine Specified Normal Distribution
Assuming a normal distribution:
- Mean (μ) = 69
- Standard Deviation (σ) = 17.8237

### Step 5: Calculate Z-scores
For each value in the sample, we calculate the Z-score using the formula:
\[ Z = \frac{X - \mu}{\sigma} \]
where:
- \( X \) is the value in the sample,
- \( \mu \) is the mean (69),
- \( \sigma \) is the standard deviation (17.8237).

### Step 6: Compute the Z-scores
Let's calculate the Z-scores for each sample point:

1. For \(X = 50\):
\[
Z = \frac{50 - 69}{17.8237} = \frac{-19}{17.8237} \approx -1.064
\]

2. For \(X = 80\):
\[
Z = \frac{80 - 69}{17.8237} = \frac{11}{17.8237} \approx 0.617
\]

3. For \(X = 90\):
\[
Z = \frac{90 - 69}{17.8237} = \frac{21}{17.8237} \approx 1.176
\]

4. For \(X = 50\):
\[
Z = -1.064
\]

5. For \(X = 55\):
\[
Z = \frac{55 - 69}{17.8237} \approx -0.785
\]

And so on for each data point.

### Step 7: Create a Frequency Table of the Z-scores
After calculating the Z-scores for all observations, we can create intervals for our frequency table and tally how many observations fall into each interval. Here we can use standard Z-score intervals (e.g., \( (-\infty, -2), (-2, -1), (-1, 0), (0, 1), (1, 2), (2, \infty) \))

### Step 8: Perform Chi-Square Goodness of Fit Test
Using the frequencies obtained and comparing them with the expected frequencies calculated from the normal distribution, we can use the Chi-Square statistic:
\[
\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}
\]
where \( O_i \) is the observed frequency in each interval and \( E_i \) is the expected frequency under the null hypothesis.

### Step 9: Determine the Critical Value
For a Chi-Square goodness of fit test:
- Degrees of freedom (df) = number of categories - 1 - number of parameters estimated.
- If we classify observations into appropriate intervals, the degrees of freedom can be calculated.

We compare the calculated χ² value with the critical value from the Chi-Square distribution table at 1% significance level.

### Step 10: Draw Conclusion
- If χ² calculated > χ² critical, then reject the null hypothesis (indicating the observations do not fit a normal distribution).
- If χ² calculated ≤ χ² critical, then we fail to reject the null hypothesis.

### Note:
For exact calculations and a complete analysis, numerical data processing (like calculating frequencies and obtaining χ²) would typically require software or a calculator. Please use statistical software or a calculator to compute these values for accurate results.

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