Question

A random sample of *n*_{1} = 10 regions in New
England gave the following violent crime rates (per million
population).

*x*_{1}: New England Crime
Rate

3.5 | 3.7 | 4.2 | 4.1 | 3.3 | 4.1 | 1.8 | 4.8 | 2.9 | 3.1 |

Another random sample of *n*_{2} = 12 regions in
the Rocky Mountain states gave the following violent crime rates
(per million population).

*x*_{2}: Rocky Mountain Crime
Rate

3.9 | 4.1 | 4.5 | 5.1 | 3.3 | 4.8 | 3.5 | 2.4 | 3.1 | 3.5 | 5.2 | 2.8 |

Assume that the crime rate distribution is approximately normal in both regions.

(i) Use a calculator to calculate *x*_{1},
*s*_{1}, *x*_{2}, and
*s*_{2}. (Round your answers to three decimal
places.)

x_{1} |
= |

s_{1} |
= |

x_{2} |
= |

s_{2} |
= |

(ii) Do the data indicate that the violent crime rate in the Rocky
Mountain region is higher than in New England? Use *α* =
0.01.

(a) What is the level of significance?

State the null and alternate hypotheses.

*H*_{0}: *μ*_{1} =
*μ*_{2}; *H*_{1}:
*μ*_{1} >
*μ*_{2}*H*_{0}:
*μ*_{1} = *μ*_{2};
*H*_{1}: *μ*_{1} ≠
*μ*_{2} *H*_{0}:
*μ*_{1} = *μ*_{2};
*H*_{1}: *μ*_{1} <
*μ*_{2}*H*_{0}:
*μ*_{1} < *μ*_{2};
*H*_{1}: *μ*_{1} =
*μ*_{2}

(b) What sampling distribution will you use? What assumptions are
you making?

The standard normal. We assume that both population
distributions are approximately normal with unknown standard
deviations.The Student's *t*. We assume that both population
distributions are approximately normal with unknown standard
deviations. The standard normal. We assume
that both population distributions are approximately normal with
known standard deviations.The Student's *t*. We assume that
both population distributions are approximately normal with known
standard deviations.

What is the value of the sample test statistic? (Test the
difference *μ*_{1} − *μ*_{2}. Round
your answer to three decimal places.)

(c) Find (or estimate) the *P*-value.

*P*-value > 0.2500.125 < *P*-value <
0.250 0.050 < *P*-value <
0.1250.025 < *P*-value < 0.0500.005 <
*P*-value < 0.025*P*-value < 0.005

Sketch the sampling distribution and show the area corresponding to
the *P*-value.

(d) Based on your answers in parts (a) to (c), will you reject or
fail to reject the null hypothesis? Are the data statistically
significant at level *α*?

At the *α* = 0.01 level, we fail to reject the null
hypothesis and conclude the data are statistically significant.At
the *α* = 0.01 level, we reject the null hypothesis and
conclude the data are not statistically
significant. At the *α* = 0.01 level,
we fail to reject the null hypothesis and conclude the data are not
statistically significant.At the *α* = 0.01 level, we reject
the null hypothesis and conclude the data are statistically
significant.

(e) Interpret your conclusion in the context of the
application.

Fail to reject the null hypothesis, there is insufficient evidence that violent crime in the Rocky Mountain region is higher than in New England.Reject the null hypothesis, there is insufficient evidence that violent crime in the Rocky Mountain region is higher than in New England. Reject the null hypothesis, there is sufficient evidence that violent crime in the Rocky Mountain region is higher than in New England.Fail to reject the null hypothesis, there is sufficient evidence that violent crime in the Rocky Mountain region is higher than in New England.

Answer #1

(i)

x1 | x2 | |

3.550 | 3.850 | mean |

0.841 | 0.909 | std. dev. |

(a) 0.01

*H*_{0}: *μ*_{1} =
*μ*_{2}; *H*_{1}:
*μ*_{1} < *μ*_{2}

(b) The Student's *t*. We assume that both population
distributions are approximately normal with unknown standard
deviations.

-0.797

(c)

0.125 < *P*-value < 0.250

(d)

At the *α* = 0.01 level, we fail to reject the null
hypothesis and conclude the data are not statistically
significant.

(e)

Fail to reject the null hypothesis, there is insufficient evidence that violent crime in the Rocky Mountain region is higher than in New England.

x1 | x2 | |

3.550 | 3.850 | mean |

0.841 | 0.909 | std. dev. |

10 | 12 | n |

20 | df | |

-0.3000 | difference (x1 - x2) | |

0.7728 | pooled variance | |

0.8791 | pooled std. dev. | |

0.3764 | standard error of difference | |

0 | hypothesized difference | |

-0.797 | t | |

.2174 | p-value (one-tailed, lower) |

A random sample of n1 = 10 regions in New
England gave the following violent crime rates (per million
population).
x1: New England Crime
Rate
3.5
3.9
4.0
4.1
3.3
4.1
1.8
4.8
2.9
3.1
Another random sample of n2 = 12 regions in
the Rocky Mountain states gave the following violent crime rates
(per million population).
x2: Rocky Mountain Crime
Rate
3.9
4.3
4.7
5.3
3.3
4.8
3.5
2.4
3.1
3.5
5.2
2.8
Assume that the crime rate distribution...

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3.1
Another random sample of n2 = 12 regions in the Rocky Mountain
states gave the following violent crime rates (per million
population).
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A random sample of n1 = 10 regions in New
England gave the following violent crime rates (per million
population).
x1: New
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3.3
3.7
4.2
4.1
3.3
4.1
1.8
4.8
2.9
3.1
Another random sample of n2 = 12 regions in
the Rocky Mountain states gave the following violent crime rates
(per million population).
x2: Rocky
Mountain Crime Rate
3.9
4.1
4.5
5.5
3.3
4.8
3.5
2.4
3.1
3.5
5.2
2.8
Assume that the crime rate distribution is approximately...

A random sample of n1 = 10 regions in New
England gave the following violent crime rates (per million
population).
x1: New
England Crime Rate
3.3
3.7
4.2
4.1
3.3
4.1
1.8
4.8
2.9
3.1
Another random sample of n2 = 12 regions in
the Rocky Mountain states gave the following violent crime rates
(per million population).
x2: Rocky
Mountain Crime Rate
3.5
4.3
4.5
5.5
3.3
4.8
3.5
2.4
3.1
3.5
5.2
2.8
Assume that the crime rate distribution is approximately...

A random sample of n1 = 10 regions in New
England gave the following violent crime rates (per million
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3.5
3.9
4.0
4.1
3.3
4.1
1.8
4.8
2.9
3.1
Another random sample of n2 = 12 regions in
the Rocky Mountain states gave the following violent crime rates
(per million population).
x2: Rocky Mountain Crime
Rate
3.7
4.1
4.7
5.3
3.3
4.8
3.5
2.4
3.1
3.5
5.2
2.8
Assume that the crime rate distribution...

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The Wind Mountain archaeological site is in southwest New
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