To test the hypothesis that the proportion of stocks that went up is significantly more than 0.3, we will conduct a one-sample z-test for proportions.
Step 1: Set up the hypotheses
- Null hypothesis (\(H_0\)): \(p = 0.3\)
- Alternative hypothesis (\(H_a\): \(p > 0.3\)
Step 2: Calculate the sample proportion
Let \( n = 56 \) (the sample size) and \( x = 24 \) (the number of stocks that went up). The sample proportion (\( \hat{p} \)) is calculated as follows:
\[ \hat{p} = \frac{x}{n} = \frac{24}{56} \approx 0.42857 \]
Step 3: Calculate the test statistic
The test statistic for a proportion is calculated using the formula:
\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \]
Where \( p_0 = 0.3 \).
- Calculate the standard error:
\[ SE = \sqrt{\frac{0.3(1 - 0.3)}{56}} = \sqrt{\frac{0.3 \times 0.7}{56}} = \sqrt{\frac{0.21}{56}} \approx \sqrt{0.00375} \approx 0.0612 \]
- Now, plug in the values to calculate the z statistic:
\[ z = \frac{0.42857 - 0.3}{0.0612} \approx \frac{0.12857}{0.0612} \approx 2.0955 \]
Test Statistic: \[ z \approx 2.096 \text{ (to three decimal places)} \]
Step 4: Calculate the p-value
For a one-tailed test, we need to find the probability that \( z \) is greater than 2.096. Using the standard normal distribution table or a calculator, we can find the p-value.
The p-value for \( z = 2.096 \):
Looking up \( z = 2.096 \):
- The cumulative probability (area to the left of z) is approximately 0.9817.
Thus, the p-value is:
\[ \text{p-value} = 1 - 0.9817 = 0.0183 \]
P-value: \[ p = 0.0183 \text{ (to four decimal places)} \]
Summary of results
- Test statistic: \( z \approx 2.096 \)
- P-value: \( p \approx 0.0183 \)
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