Your Babysitter Claims She Is Underpaid Given The Current Co

Your Babysitter Claims That She Is Underpaid Given the Current Market

Your babysitter claims that she is underpaid given the current market. Her hourly wage is $12 per hour. You do some research and discover that the average wage in your area is $14 per hour with a standard deviation of 1.9. Calculate the Z score and use the table to find the standard normal probability. Based on your findings, should you give her a raise?

Explain your reasoning as to why or why not Tutor O-rama claims that their services will raise student SAT math scores at least 50 points. The average score on the math portion of the SAT is μ = 350 and σ = 35. The 100 students who completed the tutoring program had an average score of 385 points. Is the average score of 385 points significant at the 5% level? Is it significant at the 1% level? Explain why or why not.

Paper For Above instruction

Assessment of the Babysitter’s Pay and SAT Score Significance

The decision to give a babysitter a raise based on market wages requires analyzing her current wage relative to the local distribution. Her hourly wage of $12 must be compared to the area's average wage to determine if she is underpaid. Using the provided data, the average hourly wage in the area is $14 with a standard deviation of $1.9.

The Z-score is a statistical measure that tells us how many standard deviations an individual data point is from the mean. It is calculated as:

Z = (X - μ) / σ

where X is the individual data point (her wage), μ is the mean, and σ is the standard deviation.

Plugging the given values into the formula:

Z = (12 - 14) / 1.9 ≈ -2 / 1.9 ≈ -1.05

A Z-score of approximately -1.05 indicates that her wage is 1.05 standard deviations below the average wage.

Referring to the standard normal distribution table, the probability associated with a Z-score of -1.05 is approximately 0.1464. This indicates that roughly 14.64% of wages in the area are below her current wage.

In considering whether to give her a raise, the key question is whether her wage is significantly below the average. Since a common threshold for statistical significance at the 5% level is a Z-score less than -1.645, her Z-score of -1.05 does not meet this criterion. Therefore, statistically, her wage is not significantly below the average; she is within a typical wage range in her area.

Based on these findings, I would not argue that she is underpaid to the extent that warrants a significant adjustment. However, other factors, such as her experience and performance, should also inform the decision beyond mere statistics.

Evaluating SAT Score Improvement After Tutoring

Turning to the claim by Tutor O-rama that their services can raise SAT math scores by at least 50 points, we analyze the statistical significance of the observed improvement. The population mean μ is 350, with a standard deviation σ of 35, and 100 students participated in the tutoring program, achieving an average score of 385.

To assess whether this improvement is statistically significant, we perform a hypothesis test comparing the sample mean to the population mean. The null hypothesis (H₀) states that the tutoring program does not improve scores beyond 50 points, while the alternative hypothesis (H₁) suggests that the program does improve scores by at least 50 points.

Calculating the standard error of the mean (SEM):

SEM = σ / √n = 35 / √100 = 35 / 10 = 3.5

Next, compute the Z-score for the observed sample mean:

Z = (X̄ - μ) / SEM = (385 - 350) / 3.5 = 35 / 3.5 = 10

A Z-score of 10 indicates an extremely significant difference from the population mean.

Since the question relates to whether the observed increase of 35 points (from 350 to 385) exceeds 50 points, we note that the actual increase is 35 points. Nevertheless, the statistical test tells us that this improvement is highly unlikely due to chance (p

At the 5% significance level, a Z-score of approximately 1.96 corresponds to the threshold for significance, and our Z of 10 far exceeds that, indicating the increase is highly statistically significant. Similarly, at the 1% level, the Z critical is about 2.58, and our result still exceeds this.

However, for the claim that scores increased by at least 50 points, the observed increase is only 35 points, which is below the claimed threshold. Despite the increase being statistically significant, it does not substantiate the claim of a 50-point improvement.

In conclusion, the data strongly suggests that the tutoring program does improve scores significantly; however, the magnitude of improvement is less than the claimed minimum of 50 points. The statistical evidence confirms effectiveness but does not support the specific claim of a 50-point increase.

References

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  • OpenStax College. (2015). Introductory Statistics. OpenStax CNX. https://openstax.org/details/books/introduction-statistics
  • Pearson Education. (2017). SAT Test Scores and Students' Performance. Pearson Educational Reports.
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