There is a chance that the observed sample proportion of 0.855 is due to random sampling variation and may still be consistent with the hypothesized proportion of 0.90.
To test the management's claim that the proportion of games won when scoring 102 or more points is 0.90, we can conduct a one-proportion z-test. Here's a step-by-step explanation of the process:
State the Hypotheses:
Null Hypothesis (H₀): p = 0.90 (The proportion of games won is 0.90)
Alternative Hypothesis (H₁): p ≠ 0.90 (The proportion of games won is not 0.90)
Collect Data:
Suppose you randomly sample 200 games from the team's records and find that 171 of those games were won when scoring 102 or more points. This means the sample proportion, denoted by "p", is 171/200 = 0.855.
Calculate the Test Statistic (z-score):
z = (p - p₀) / √(p₀(1 - p₀)/n)
z = (0.855 - 0.90) / √(0.90(1 - 0.90)/200)
z ≈ -1.77
Determine the Critical Value:
Since the alternative hypothesis is two-tailed (≠), we need to find the critical values for both tails of the z-distribution at a significance level of 0.05. Using a z-table or calculator, we find the critical values to be -1.96 and 1.96.
Compare the Test Statistic to the Critical Values:
The calculated z-score (-1.77) falls within the critical region (-1.96 < -1.77 < 1.96), indicating that we fail to reject the null hypothesis.
Interpret the Results:
At a 5% level of significance, we do not have sufficient evidence to reject the management's claim that the proportion of games won when scoring 102 or more points is 0.90. There is a chance that the observed sample proportion of 0.855 is due to random sampling variation and may still be consistent with the hypothesized proportion of 0.90.