Question

Baseball has always been a favorite pastime in America and is rife with statistics and theories. In a paper, researchers showed that major league players who have nicknames live an average of 2½ years longer than those without them (The Wall Street Journal, July 16, 2009). You do not believe in this result and decide to collect data on the lifespan of 30 baseball players along with a nickname variable that equals 1 if the player had a nickname and 0 otherwise. The data are shown in the accompanying table and are contained in the accompanying Excel file. (You may find it useful to reference the appropriate table: z table or t table)

Years	Nickname		Years	Nickname		Years	Nickname
74	1		61	0		68	1
62	1		64	0		68	0
67	1		70	0		64	1
73	1		71	1		67	1
49	1		69	1		64	0
62	0		56	0		63	1
56	0		68	1		68	1
63	0		70	1		68	1
80	1		79	1		74	0
65	1		67	0		64	0

Click here for the Excel Data File

Let Samples 1 and 2 represent major-league players with and without nicknames, respectively.

a. Create two subsamples consisting of players with and without nicknames. Calculate the average longevity for each subsample. (Round your answers to 4 decimal places.)

b. Specify the hypotheses to contradict the claim made by researchers.

• H₀: μ₁ − μ₂ = 2.5; H_A: μ₁ − μ₂ ≠ 2.5

• H₀: μ₁ − μ₂ ≥ 2.5; H_A: μ₁ − μ₂ < 2.5

• H₀: μ₁ − μ₂ ≤ 2.5; H_A: μ₁ − μ₂ > 2.5

c-1. Calculate the value of the test statistic. Assume that the population variances are unknown but equal. (Round your answer to 3 decimal places.)

c-2. Find the p-value.

• p-value ≤ 0.01

• 0.01 < p-value ≤ 0.02

• 0.02 < p-value ≤ 0.05

• 0.05 < p-value ≤ 0.10

• p-value > 0.10

d. What is the conclusion of the test using a 5% level of significance?

• Reject H₀; the sample data disproves the claim by the researchers.

• Reject H₀; the sample data does not disprove the claim by the researchers.

• Do not reject H₀; the sample data disproves the claim by the researchers.

• Do not reject H₀; the sample data does not disprove the claim by the researchers.

Answer 1

• The subsamples are created as:

  With Nickname
  74
  62
  67
  73
  49
  80
  65
  71
  68
  70
  79
  68
  64
  67
  63
  68
  68

  
  The mean is
  

  Without Nickname
  62
  56
  63
  61
  64
  70
  56
  67
  68
  64
  74
  64

  
  
  The mean is
  
• Here, we are to test if the player with nicknames live 2.5 years longer than those who do not.
  Hence, we are to if this claim is False. That is, if they do not live more than 2.5 years. That is, our hypothesis is:
  
  
• The test statistic is given by:
  
  Under equal variance assumption, pooled sample standard deviation is given as:
  
  where,
  
  
  Hence,
  
  
  
  Hence,
  
  
  
  Hence,
  
  
• Under Null,
  
  
  p-value:
  
  
  
  Hence, the p-value>0.1
  
  
• Hence, the Null is rejected at 10% level of significance, and hence we conclude that players who have nicknames live an average of 2½ years longer than those without them.
  
  Hence, the correct option is:
  Do not reject H₀; the sample data does not disprove the claim by the researchers.
  
  

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