You wish to test the following claim (H a) at a significance level of alpha =0.01

Question

.
H_o : p = 0.4
H_a : p < 0.4

You obtain a sample of size n=212 in which there are 76 successful observations. For this test, you should NOT use the continuity correction, and you should use the normal distribution as an approximation for the binomial distribution.

What is the test statistic for this sample? (Report answer accurate to three decimal places.)
test statistic =

Answer 1

To calculate the test statistic for the given hypothesis test, we will use the formula for a z-test for proportions.

1. Define the parameters:

   • \( n = 212 \) (sample size)
   • \( x = 76 \) (number of successes)
   • \( p_0 = 0.4 \) (hypothesized population proportion under \( H_0 \))

2. Calculate the sample proportion: \[ \hat{p} = \frac{x}{n} = \frac{76}{212} \approx 0.3585 \]

3. Calculate the standard error of the sampling distribution of the proportion: \[ SE = \sqrt{\frac{p_0(1 - p_0)}{n}} = \sqrt{\frac{0.4 \times (1 - 0.4)}{212}} = \sqrt{\frac{0.4 \times 0.6}{212}} = \sqrt{\frac{0.24}{212}} \approx \sqrt{0.001132} \approx 0.0336 \]

4. Calculate the z-test statistic: \[ z = \frac{\hat{p} - p_0}{SE} = \frac{0.3585 - 0.4}{0.0336} \approx \frac{-0.0415}{0.0336} \approx -1.237 \]

5. Report the test statistic accurate to three decimal places: \[ \text{test statistic} \approx -1.237 \]

Thus, the final answer for the test statistic is: \[ \text{test statistic} = -1.237 \]