Question

The data show the number of hits and the number of at bats for 7 major league players in recent world series, Is there a linear relationship between the number of hits a world series player gets and the number of times at bat the player has? Find y' when x=60.

at bats 51 67 77 44 55 39 45

Hits 19 25 30 20 23 16 18

a. compute the value of the correlation coefficient

b. state the hypotheses

c. test the significance of the correlation coefficient at α=0.05 using table 1

d. give a brief explanation of the tyoe of relationship assume all assumptions have been met

Answer 1

Here we're given the data on "the no. of hits a world series player gets" and "the number of times at bat the player has"

at bats(x)	Hits(y)
51	19
67	25
77	30
44	20
55	23
39	16
45	18

To check whether, there is a Linear Relationship between "the no. of hits a world series player gets" and "the number of times at bat the player has" we've to plot the data on the graph.

Graph-

We draw this graph using "Minitab'16"

Steps:

1. Enter the data in "Minitab'16"
2. Click on "Stat"
3. Select "Regression"
4. Choose "Fitted Line Plot"
5. Choose "at bats" as "Predictor" and "Hits" as "Response".
6. Choose "Linear" as "Type of Regression Model"
7. Click on "OK"

Conclusion: from the above graph we can say that the relationship between  "the no. of hits a world series player gets" and "the number of times at bat the player has" is "Linear".

In order to get the value of y' when x=60 we've to fit a Liner Regression Model

Suppose,

where, and   

The fitted Regression Equation is-

when, x=60,  

a) Correlation Coefficient-

b) Hypothesis to be tested-

Vs   

where,   Population Correlation Coefficient

c) Test of Significance of the Correlation Coefficient-

• Test Statistics-
  
  

where, r= sample correlation coeffocient
n= sample size

• Rejection Criteria:
  We reject the null hypothesis at level of significance iff,
  
  
  where, is the percentile value of the "t-distribution" with n-2 degrees of freedom.
  
  here, and n=7
  So,
  
• Decision:
  
  As, so we can conclude on the basis of the given data, that we reject the Null the Hypothesis at 0.05 level of significance and r is significant.

​​​​​​​d) Explanation:

There is a Linear Relation between "at bats"(x) and "Hits"(y). i.e., if x increase at the rate of 0.3402 y also increases at the same rate.

Assumptions:

The residuals are normally distributed with zero mean and variance i.e,

I hope this clarifies your doubt. If you're satisfied with the solution, hit the Like button. For further clarification, comment below. Thank You. :)