It is known from the records of the City Schools that the standard deviation
of Algebra scores is 6.0. A sample of 200 students from the system was
taken and was found out that the sample mean score is 75. Previous tests
showed the population mean to be 70. Is it safe to conclude that the
sample is significantly different from the population? Use α= 0.05.
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To determine if the sample is significantly different from the population, we need to perform a hypothesis test. In this case, we are comparing the sample mean to the population mean.
Step 1: State the null and alternative hypotheses.
The null hypothesis (H0) assumes no significant difference between the sample mean and the population mean. The alternative hypothesis (Ha) assumes that there is a significant difference.
H0: μ = 70 (population mean is equal to 70)
Ha: μ ≠ 70 (population mean is not equal to 70)
Step 2: Determine the significance level (α).
The significance level, denoted by α, is the probability of rejecting the null hypothesis when it is true. In this case, α is given as 0.05.
Step 3: Calculate the test statistic.
To calculate the test statistic, we will use the formula for the z-test:
z = (x̄ - μ) / (σ / √n)
Where:
x̄ = sample mean (75)
μ = population mean (70)
σ = standard deviation (6.0)
n = sample size (200)
Plugging in the values, we get:
z = (75 - 70) / (6.0 / √200)
Step 4: Determine the critical value(s).
Since we have a two-tailed test (Ha: μ ≠ 70), we need to divide the significance level by 2 to get the critical values for both tails. Using a z-table or a statistical software, we can determine the critical z-values.
For α = 0.05, the critical z-values are approximately ±1.96.
Step 5: Make a decision.
If the calculated test statistic falls within the critical z-values, we fail to reject the null hypothesis. If it falls outside the critical values, we reject the null hypothesis and conclude that there is a significant difference between the sample mean and the population mean.
Step 6: Calculate the p-value (optional).
If desired, we can calculate the p-value associated with the test statistic. The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated. We compare the p-value to the significance level to make a decision.
Step 7: State the conclusion.
Based on the calculated test statistic and the critical values, we compare them to make a decision. If the test statistic falls outside the critical values (beyond ±1.96), we reject the null hypothesis. If it falls within the critical values, we fail to reject the null hypothesis.
In this case, calculate the test statistic (z) using the given values. Compare the calculated z-value to the critical values of ±1.96. If the calculated z-value falls outside this range, we reject the null hypothesis. If it falls within the range, we fail to reject the null hypothesis.
Note: Since we have not been given the actual calculated value of z, you will need to calculate it using the provided formula.