Tutor O Rama Claims That Their Services Will Raise StudentSc

Tutor O Rama Claims That Their Services Will Raise Student Sat Math Sc

Button O Rama asserts that their tutoring program will enhance students' SAT math scores by a minimum of 50 points. The average score on the math section of the SAT is 350, with a standard deviation of 35. In a sample of 100 students who completed the program, the mean score was 385. The question is whether this observed average is statistically significant at the 5% level and at the 1% level. Clarify whether these results are statistically meaningful at each of these significance thresholds and explain the reasoning behind your conclusion.

Paper For Above instruction

In assessing whether Tutor O Rama's claim about improving SAT math scores holds statistical validity, we apply hypothesis testing principles to interpret the observed data. The central question is whether the observed mean score of 385 points among 100 students constitutes a significant increase over the average of 350 points, considering the population's standard deviation of 35 points. This requires calculating the z-score and comparing it against critical values corresponding to the 5% and 1% significance levels.

The null hypothesis (H₀) posits that Tutor O Rama's program has no effect, meaning the mean score remains at the population average of 350. The alternative hypothesis (H₁) suggests that the program results in an increase of at least 50 points, implying the true mean score exceeds 350 by a significant margin. In this context, since the claim involves a minimum increase, a one-sided test is appropriate.

Calculating the standard error (SE) of the mean involves dividing the population standard deviation by the square root of the sample size: SE = σ / √n = 35 / √100 = 35 / 10 = 3.5. The z-score then measures how many standard errors the sample mean (385) is above the population mean (350):

z = (X̄ - μ) / SE = (385 - 350) / 3.5 = 35 / 3.5 = 10.

A z-score of 10 is exceptionally high and far exceeds the critical z-values typically associated with significance levels of 0.05 (approximately 1.645) and 0.01 (approximately 2.33). Since 10 is much larger than these thresholds, the result is statistically significant at both the 5% and the 1% levels.

This indicates that the observed increase in scores is highly unlikely to be due to random chance alone, providing strong evidence to support Tutor O Rama's claim. The substantial z-score confirms that the program's effect on scores is statistically meaningful and unlikely to be attributable to variability within the population.

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