A training center claims that 75% of people taking national exam on achievement. In a sample of 90 people who took the exam only 39 have passed the exam. Use hypothesis testing to determine whether the training center claim is justified. Use 5% level of Significance
You can use a one-sample proportional z-test for your data. (Test sample proportion = 39/90 and sample size = 90) Convert fractions to decimals to use in the z-test. Find the critical value in the appropriate table at .05 level of significance for a two-tailed test. Compare the test statistic you calculate to the critical value from the table. If the test statistic exceeds the critical value, reject the null. If the test statistic does not exceed the critical value, do not reject the null. You can draw your conclusions from there.
To determine whether the training center's claim is justified, we can conduct a hypothesis test. The null hypothesis (H0) is that the true proportion of people who pass the exam is 75%, while the alternative hypothesis (H1) is that the true proportion is different from 75%.
Step 1: Formulate the hypotheses:
H0: p = 0.75 (The true proportion of people who pass the exam is 75%)
H1: p ≠ 0.75 (The true proportion is different from 75%)
Step 2: Determine the significance level:
The significance level given in this case is 5% or 0.05.
Step 3: Calculate the test statistic:
We will use the standard deviation formula for proportions. The test statistic formula for this case is:
z = (p̂ - p0) / √(p0 * (1 - p0) / n),
where:
p̂ is the sample proportion (39/90),
p0 is the assumed proportion under the null hypothesis (0.75),
n is the sample size (90).
Plugging in the values:
z = (39/90 - 0.75) / √(0.75 * (1 - 0.75) / 90)
Step 4: Calculate the p-value:
The p-value is the probability of obtaining a test statistic as extreme as the one calculated, assuming the null hypothesis is true. We can find the p-value using a standard normal distribution table or calculator.
Step 5: Make a decision:
If the p-value is less than the significance level (0.05), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
So, now you need to calculate the z-value and find the corresponding p-value from a standard normal distribution table or use statistical software to determine the exact p-value. If the p-value is less than 0.05, reject the null hypothesis and conclude that the training center's claim is not justified.