Question

**NBA players:** Michael wants to investigate
whether the distribution of his classmates’ favorite NBA players of
all time is different from the one reported by NBA. He randomly
surveys his classmates. The table below contains the percentages
reported by NBA and the observed counts from his random sample.

Player | LeBron James | Kobe Bryant | Steph Curry | Michael Jordan | Others |

NBA % | 20% | 30% | 15% | 19% | 16% |

Observed Counts | 23 | 28 | 10 | 15 | 11 |

a. If the percentages are as reported by NBA, then what is the
degrees of freedom for the test statistic?

b. If the percentages are as reported by NBA, then what is the p-value for this hypothesis test?

c. Michael wants to carry out a statistical inference procedure for this scenario using the χ2 -distribution. Which of the following conditions must be met? (Select all that apply)

- The difference in all calculated proportions must be at least 5.

- The sample size must be at least 30 or the population data must be normally distributed.

- There must be at least 10 successes and 10 failures for each level of the categorical variable.

- The observations must be independent.

- There must be an expected count of at least 5 for each level of the categorical variable.

- There must be at least 3 levels of the categorical variable.

Answer #1

Player |
Observed Count |
Expected Proportion |
Expected Count |
(O-E)^2/E |

LeBron James | 23 | 0.2 | 0.2*87 = 17.40 | 1.8023 |

Kobe Bryant | 28 | 0.3 | 0.3*87 = 26.10 | 0.1383 |

Steph Curry | 10 | 0.15 | 0.15*87 = 13.05 | 0.7128 |

Michael Jordan | 15 | 0.19 | 0.19*87 = 16.53 | 0.1416 |

Others | 11 | 0.16 | 0.16*87 = 13.92 | 0.6125 |

Total Count = | 87 | χ2 = Σ(O-E)^2/E = | 3.4076 |

Degrees of Freedom = n-1 = 5-1 = **4**

P-value = **0.4921** [Obtained using MS Excel
function =CHISQ.DIST.RT(3.4076,4)]

The following conditions must be met:

The observations must be independent.

There must be an expected count of at least 5 for each level of the categorical variable.

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