Question

The Nero Match Company sells matchboxes that are supposed to
have an average of 40 matches per box, with *σ* = 8. A
random sample of 96 matchboxes shows the average number of matches
per box to be 42.5. Using a 1% level of significance, can you say
that the average number of matches per box is more than 40?

What are we testing in this problem?

**A.)** single mean

**B.)** single
proportion

(a) What is the level of significance?

State the null and alternate hypotheses.

**A.)** *H*_{0}: *p* = 40;
*H*_{1}: *p* > 40

**B.)** H_{0}: *μ* = 40;
*H*_{1}: *μ* <
40

**C.)** *H*_{0}: *μ* = 40;
*H*_{1}: *μ* ≠ 40

**D.)** *H*_{0}: *p* = 40;
*H*_{1}: *p* < 40

**E.)** *H*_{0}: *μ* = 40;
*H*_{1}: *μ* > 40

**F.)** *H*_{0}: *p* = 40;
*H*_{1}: *p* ≠ 40

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

**A.)**The standard normal, since we assume that
*x* has a normal distribution with known *σ*.

**B.)**The Student's *t*, since we assume
that *x* has a normal distribution with known
*σ*.

**C.)**The standard normal, since we assume that
*x* has a normal distribution with unknown *σ*.

**D.)**The Student's *t*, since we assume
that *x* has a normal distribution with unknown
*σ*.

What is the value of the sample test statistic? (Round your answer
to two decimal places.)

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

**A.)***P*-value > 0.250

**B.)**0.125 < *P*-value <
0.250

**C.)**0.050 < *P*-value < 0.125

**D.)**0.025 < *P*-value < 0.050

**E.)**0.005 < *P*-value < 0.025

**F.)***P*-value < 0.005

(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*?

**A.)**At the *α* = 0.01 level, we reject
the null hypothesis and conclude the data are statistically
significant

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

**C.)**At the *α* = 0.01 level, we fail to
reject the null hypothesis and conclude the data are statistically
significant.

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

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

**A.)**There is sufficient evidence at the 0.01
level to conclude that the average number of matches per box is now
greater than 40.

**B.)**There is insufficient evidence at the 0.01
level to conclude that the average number of matches per box is now
greater than 40.

Answer #1

**A.)** single mean

a) level of significance=0.01

**A.)** *H*_{0}: *p* = 40;
*H*_{1}: *p* > 40

**A.)**The standard normal, since we assume that
*x* has a normal distribution with known *σ*.

population mean μ= | 40 |

sample mean 'x̄= | 42.500 |

sample size n= | 96.00 |

std deviation σ= | 8.000 |

std error ='σx=σ/√n= | 0.8165 |

test stat z = '(x̄-μ)*√n/σ= |
3.06 |

c)

p value = | 0.0011 |

**F.)***P*-value < 0.005

d)**A.)**At the *α* = 0.01 level, we reject
the null hypothesis and conclude the data are statistically
significant

e)

**A.)**There is sufficient evidence at the 0.01
level to conclude that the average number of matches per box is now
greater than 40.

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