Need Solutions And Calculations, Also Excel Solution For Eac

Need Solutions Calculations But Also Excel Solutionq1for Each Of The

Need solutions calculations but also excel solution Q1. For each of the following claims mention what type of test should be used (one-tail, two-tail) and set up the null and alternate hypothesis. Consider that the critical value is = CV (like for example 1.96). For each H0, mention the acceptance range.

a) The 87Cents Store claims that their daily profit is at least $5000.

b) The Pastry shop claims that its chocolate cookies have no more than 50% chocolate per cookie.

c) PS193 claims that its students scored in the 95thile.

Paper For Above instruction

The following is an analysis and solution to the statistical claims provided, including the appropriate tests, hypotheses, and statistical calculations.

Question 1: Testing Claims with Hypotheses and Tests

a) The 87Cents Store claims that their daily profit is at least $5000.

Type of test: One-tailed (upper tail), since the claim is that profit is at least $5000, which is a greater than or equal to statement. We will test if the mean profit is significantly less than $5000.

Null hypothesis (H0): μ ≥ 5000

Alternative hypothesis (H1): μ

Acceptance Range: For a significance level α (not specified, but typically 0.05), the critical value CV is 1.96 for a two-tailed test; for a one-tailed test, the critical z-value is approximately 1.645. If the calculated z-value is less than -1.645, we reject H0.

b) The Pastry shop claims that its chocolate cookies have no more than 50% chocolate per cookie.

Type of test: One-tailed (upper tail), testing if the true mean percentage exceeds 50%.

Null hypothesis (H0): p ≤ 50%

Alternative hypothesis (H1): p > 50%

Acceptance Range: Using the z-test for proportions, with α = 0.05, the critical z-value is 1.645. If the computed z exceeds 1.645, reject H0.

c) PS193 claims that its students scored in the 95th percentile.

Since the claim pertains to a specific percentile, hypothesis testing involves whether the proportion of students scoring above a certain cutoff matches the 95th percentile. This is typically a proportion test.

Null hypothesis (H0): The proportion of students scoring above the cutoff equals 5% (since 95th percentile).

Alternative hypothesis (H1): The proportion differs from 5% (two-tailed), or specifically exceeds/less than that, depending on direction.

Question 2: Testing the Average Tuition with Sample Data

A researcher wants to test whether the average tuition is at least $7000. Using a sample of 81 colleges, mean = $6900, standard deviation = $500, at α = 0.06.

a) Test of the claim

Null hypothesis (H0): μ ≥ 7000

Alternative hypothesis (H1): μ

Calculate the test statistic (z):

z = (sample mean - hypothesized mean) / (standard deviation / √n) = (6900 - 7000) / (500 / √81) = (-100) / (500 / 9) = (-100) / 55.56 ≈ -1.80

Critical z-value at α=0.06 (one-tailed) is approximately -1.55. Since -1.80

b) Confidence interval construction

Construct a 95% two-sided confidence interval for the mean.

Margin of error (ME) = z(α/2) (σ/√n) = 1.96 (500/9) ≈ 1.96 55.56 ≈ 108.88

CI: (6900 - 108.88, 6900 + 108.88) = (6791.12, 7008.88)

The CI suggests that the true mean tuition could be as low as approximately $6791 or as high as $7009, supporting the hypothesis that the average is close to or below $7000.

Question 3: Salary of Substitute Teachers in New York

The claim is that the average salary is at most $100 per day. A random sample of ten districts is taken, the sample mean and standard deviation are calculated, and a hypothesis test is conducted at α=0.05.

Hypotheses:

H0: μ ≥ 100

H1: μ

Suppose the sample data yields a mean x̄ and standard deviation s; the t-test is appropriate here because the sample size is small (

t = (x̄ - 100) / (s / √n)

Compare the calculated t-value with the critical t-value for 9 degrees of freedom at α=0.05 (approximately 1.833). If t

Question 4: Short answer questions and multiple choice questions

1. Confidence Interval Explanation

A 90% confidence interval estimating that 49% ± 4% of customers buy at least two items suggests that we are 90% confident that the true proportion of such customers is between 45% and 53%. The interval reflects the precision of the estimate based on the sample size of 100 customers.

2. Range of Correlation Coefficient and Interpretation

The correlation coefficient r ranges from -1 to 1, indicating perfect negative to perfect positive linear relationships. An r value of -0.78 indicates a strong negative correlation, meaning as one variable increases, the other tends to decrease.

Question 5: Regression Analysis Explanation

The regression output provides key statistics: R (correlation coefficient), p-values for the coefficients, and F-statistic. The R value measures the strength of the linear relationship; the p-value indicates whether the relationship is statistically significant; the F-value tests the overall significance of the regression model. The regression equation is derived from the slope and intercept coefficients, typically in the form Y = a + bX.

Question 6: Alumni Contributions Analysis

a) Using Excel, plot a scatter plot of years since graduation (X) versus contribution (Y). The plot visually reveals the data trend, whether it suggests a positive, negative, or no relationship.

b) The correlation coefficient (r) can be calculated using Excel functions, indicating the strength and direction of the linear relationship.

c) The coefficient of determination (r²) explains the proportion of variance in contribution explained by the years since graduation.

d) The regression coefficients (intercept and slope) are calculated via Excel's regression analysis tool; the regression equation models contribution as a function of years after graduation.

e) The r value indicates the strength and direction of the linear relationship, r² shows the percentage of variance explained, and the regression equation predicts contribution at specific years. For example, at x=0 (immediately after graduation), the predicted contribution is the intercept value.

f) Using the regression equation, estimate contribution for a student who graduated 7 years ago. The point at which the contribution becomes zero or negative indicates when alumni stop contributing significantly.

References

  • Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. Sage Publications.
  • Gravetter, F. J., & Wallnau, L. B. (2017). Statistics for Behavioral Science. Cengage Learning.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the Practice of Statistics. W. H. Freeman.
  • Triola, M. F. (2018). Elementary Statistics. Pearson.
  • Ott, R. L., & Longnecker, M. (2015). An Introduction to Statistical Methods and Data Analysis. Cengage Learning.
  • Newbold, P., Carlson, W. L., & Thorne, B. (2013). Statistics for Business and Economics. Pearson.
  • Altman, D. G., & Royston, P. (2006). The cost of dichotomising continuous variables. BMJ, 332(7549), 1080.
  • Higgins, J. P., & Green, S. (2011). Cochrane Handbook for Systematic Reviews of Interventions. The Cochrane Collaboration.
  • Larson, R., & Verhulst, F. C. (1999). The dynamics of adolescent development and well-being: An overview. Journal of Youth and Adolescence.
  • Tabachnick, B., & Fidell, L. (2013). Using Multivariate Statistics. Pearson.

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