Problem 10CRE

Refer to the following list of numbers of years that U.S. presidents, popes, and British monarchs lived after their inauguration, election, or coronation, respectively. (As of this writing the last president is Gerald Ford, the last pope is John Paul II, and the last British monarch is George VI.) Assume that the data are samples randomly selected from larger populations.

Lottery: Goodness-of-Fit The bars in the histogram included with Exercise depict these frequencies: 21, 19, 15, 18, 24, 18, 16, 24, 30, and 15- Test the claim that the digits are selected from a population in which the digits are all equally likely. Is there a problem with the lottery?

Exercise

Refer to the following list of numbers of years that U.S. presidents, popes, and British monarchs lived after their inauguration, election, or coronation, respectively. (As of this writing the last president is Gerald Ford, the last pope is John Paul II, and the last British monarch is George VI.) Assume that the data are samples randomly selected from larger populations.

Lottery: Interpreting a Graph Shown below is a histogram of digits selected in California’s Win 4 lottery. Each drawing involves the random selection (with replacement) of four digits between 0 and 9 inclusive.

a. If the lottery works correctly, what should be the shape of the histogram in the long run? Does the histogram shown here depict the expected distribution?

b. Does the display depict a normal distribution? Why or why not?

Answer:

Step 1 of 2</p>

The hypotheses can be written as

H0 : The digits are selected from a population in which the digits are all equally likely

H1 : At least one of the digits selected from a population in which the digits are not equally likely.

Use a 0.05 significance level

SL. No. |
O |
p = 1/10 |
E = np |
(O - E )2 |
(O - E)2/E |

1 |
21 |
0.1 |
20 |
1 |
0.05 |

2 |
19 |
0.1 |
20 |
1 |
0.05 |

3 |
15 |
0.1 |
20 |
25 |
1.25 |

4 |
18 |
0.1 |
20 |
4 |
0.2 |

5 |
24 |
0.1 |
20 |
16 |
0.8 |

6 |
18 |
0.1 |
20 |
4 |
0.2 |

7 |
16 |
0.1 |
20 |
16 |
0.8 |

8 |
24 |
0.1 |
20 |
16 |
0.8 |

9 |
30 |
0.1 |
20 |
100 |
5 |

10 |
15 |
0.1 |
20 |
25 |
1.25 |

Sum |
200 |
1 |
200 |
208 |
10.4 |

Expected frequency (E) = np

= 200(1/10)

= 20

The calculation continues as follows. Letting E be the expected frequency of an outcome and O be the observed frequency of that outcome.