Unit Project 4(Simple Regression) Variables Dependent variable: Final data (y-axis) Independent Variable: First half x2 (x-axis)

Source of data

(Input your data source)

Regression results

Linear graph of final against first half

 .

Coefficients

Standard Error

t Stat

P-value

Intercept

-0.396408932

3.685410309

-0.1075617

0.915237539

0

0.5780696

0.110802072

5.21713712

2.40038E-05

Coefficients

Intercept

Independent Variable

Correlation coefficient

0.3964089

0.5780696

0.728985248

Standard error

Intercept

Independent Variable

Correlation coefficient

3.685410309

0.110802072

11.88148308

Z-statistic

Z statistics based on fisher’s transformation of coefficient r is

On a

Therefore the valueon a standard normal distribution, from the above p-value, the z statistics is 0.000024.

Showing where zero and one fall on the normal distribution provided, when the estimate µ, is the expected value of the coefficient.

If µ is the expected value of coefficient, using,

For 0: and for 1: this makes both 0 and 1 to lie on. That is 1 lies between 1 and 2, while 0 lies between -1and -2

Is the coefficient sufficiently different from zero? Explain.

We use the correlation coefficient value provided in question 4.

The calculated value of is greater than 3.725, the upper 0.05% point of distribution. So we reject Ho at 0.05% level and say that the coefficient is sufficiently different and greater than zero. Our coefficient displays a positive correlation between final scores and first half performance.

Is the coefficient sufficiently different from one? Explain.

Using fisher transformation of r, our hypothesis is

This has a normal distribution of

The value of 4.539 is greater than 4.4172 the upper 0.0005% point of a standard normal distribution, therefore reject Ho and say that the coefficient is sufficiently different from 1.

Project 5(Hot Hand)

Fill in the following

prob(hit/3 misses)

prob(hit/2 misses)

p(hit/1 miss)

p(hit)

p(hit/1hit)

p(hit/2 hits

p(hit/3 hits)

ρ

0.8333

0.7

0.58333

0.52

0.4076

0.5

0.5

-0.08606

Do your conditional probabilities show evidence of the hot hand? Explain.

There is no evidence of hot hand. Interpreting the above conditional probabilities, a player has higher percentage of making a hit after making a miss and in comparison a player has a lower chance of making a hit after making a hit. Evidently, there is no shooting streak since probability if making a hit after three misses is 83.3% which is higher than the probability of making a hit after three hits with a probability of 50%.

Does your correlation coefficient ρ show evidence of the hot hand? Explain.

The calculated correlation coefficient is negative. This therefore does not show evidence of the hot hand. Given a player has made one or two hits. their opponents will usually try to improve their defensive mechanism on the particular player and therefore take away the players successive shooting streaks. Defensive strategies can be the probable cause for the negative correlation between successive hits.

Fill in the following.

Wald-Wolfowitz runs test results is presented below,

Probability of observing 50 or fewer runs while 47.5 were expected (standard deviation (50) = 4.7664)

hits

misses

runs

expected runs

z-statistic

52

48

50

47.5

0.5312

Show where your runs number falls on the distribution

Z value is 0.5312, use extrapolation method to approximate the percentage value and then refer to the distribution above.

Z value of 0.5224 has a percentage of 30%, while a Z value of 0.6745 has a percentage 25%.

Divide by 2 since this is a two tailed distribution,

The runs number falls in the13.6% distribution section. This value lies between 1 to 2 and -2 to 1 section.

Is this evidence of the hot hand? Explain.

The evidence suggests existence of hot hand. The Z statistic reported above test the significance of the variation between the expected and the observed number of runs. There is always a significant difference between the expected value and observed value for individual players. Run tests performed on each player within individual games reveal compelling results.

Considering the 76ers and their opponents, data obtained of 727 basketball players show game records of more than two runs. When the observed number of runs and expected number of runs are compared there is no sufficient evidence to provide any basis for rejecting the hot hand hypothesis. Clearly 13.55% calculated is less than 13.6% tabulated therefore we accept the null hypothesis.

References

Reifman, Alan. Hot Hand: The Statistics Behind Sports Greatest Streaks. Washington, D.C: Potomac Books, 2012. Print.