In an effort to overcome the noise problem presented by his company’s impact tools (recall that μ = 89 dB) the manufacturer’s sale representative Art Snakeoil now suggests replacing the old line of noisy impact tools with his new line of quieter pistol grip impulse tools. Unfortunately, he does not

Question

In an effort to overcome the noise problem presented by his company’s impact tools (recall that μ = 89 dB) the manufacturer’s sale representative Art Snakeoil now suggests replacing the old line of noisy impact tools with his new line of quieter pistol grip impulse tools. Unfortunately, he does not have many samples to provide for testing. Nine tools are tested with the following data provided as follows (in units of dB): 85.1, 90.1, 86.3, 85.4, 89.3, 88.4, 86.3, 89.4, and 86.2.
The Purchasing Department now asks the Engineering Department to determine if the sound level of the pulse tool samples is significantly reduced from that of the impact tools presently being used.

The Engineering Department, using a significance level of α = 0.01 analyzes the noise level of the sample of nine tools by following the format:

a. State the appropriate hypothesis
b. Test the hypothesis, using α = 0.01
c. Construct a 99% confidence interval on the tool noise level.

Accepted Answer

1. [latex]H_{0}[/latex]: μ = 89
2. [latex]H_{a}[/latex]: μ < 89
3. α = 0.01 (level of significance)
4. n is small (n < 30, n = 9)
5. σ is unknown ⇒ use the t test
6. n = 9
7. From the sample data, we find

[latex]\overline{x}=87.39[/latex]

S = 1.91

From Table 4.3 for α = 0.01, n = 9, we obtain

∂ = n − 1 = 8

and

[latex]t_{0.01,8} = 2.896[/latex]

Also we find

[latex]t_{0}=\frac{\overline{x}-\mu _{0} }{S/\sqrt{n} } =\frac{87.39-89}{1.91/\sqrt{9} } =-2.53[/latex]

Displaying this data on a sketch reveals

Hence, since the value of −2.53 lies within the bounds of −2.896, we accept [latex]H_{0}[/latex], and therefore reject [latex]H_{A}[/latex] which indicates that the new (albeit small) sample of tools, at a significance level of 0.01, is statistically similar to the original noisy tools with respect to noise. Evaluating the same problem by examining its L.L. yields:

[latex]LL=\mu _{0}-t_{0.01,8}\frac{S}{\sqrt{n} }=89-2.896\frac{\left(1.91 ight) }{\sqrt{9} } \LL=87.16[/latex]

Displaying the data on a sketch yields

8. Since [latex]\overline{x} [/latex]= 87.39 > LL = 87.16
Accept [latex]H_{0}[/latex]: μ = 89
Reject [latex]H_{0}[/latex]: μ < 89

We conclude, therefore, that the means of the population from which the nine samples were randomly taken (i.e., the pulse tool samples) will have no improvement in the noise attenuation and is therefore no different or better statistically than the noise level of the original population of noisy impact tools.

TABLE 4.3
Percentile Values for Student’s t Distribution [5]

1	0.325	0.727	1.376	3.078	6.314	12.706	31.821	63.657
2	0.289	0.617	1.061	1.886	2.92	4.303	6.965	9.925
3	0.277	0.584	0.978	1.648	2.353	3.182	4.541	5.841
4	0.271	0.569	0.941	1.533	2.132	2.776	3.747	4.604
5	0.267	0.559	0.920	1.476	2.015	2.571	3.365	4.032
6	0.265	0.553	0.906	1.440	1.943	2.447	3.143	3.707
7	0.263	0.549	0.896	1.415	1.895	2.365	2.998	3.499
8	0.262	0.546	0.889	1.397	1.860	2.306	2.896	3.355
9	0.261	0.543	0.883	1.383	1.833	2.262	2.821	3.250
10	0.260	0.542	0.879	1.372	1.812	2.228	2.764	3.169
11	0.260	0.540	0.876	1.363	1.796	2.201	2.718	3.106
12	0.259	0.539	0.873	1.356	1.782	2.179	2.681	3.055
13	0.259	0.538	0.870	1.350	1.771	2.160	2.650	3.012
14	0.258	0.537	0.868	1.345	1.761	2.145	2.624	2.977
15	0.258	0.536	0.866	1.341	1.753	2.131	2.602	2.947
16	0.258	0.535	0.865	1.337	1.746	2.120	2.583	2.921
17	0.257	0.534	0.863	1.333	1.74	2.110	2.567	2.898
18	0.257	0.534	0.862	1.330	1.734	2.101	2.552	2.878
19	0.257	0.533	0.861	1.328	1.729	2.093	2.539	2.861
20	0.257	0.533	0.860	1.325	1.725	2.086	2.528	2.845
21	0.257	0.532	0.859	1.323	1.721	2.080	2.518	2.831
22	0.256	0.532	0.858	1.321	1.717	2.074	2.508	2.819
23	0.256	0.532	0.858	1.319	1.714	2.069	2.500	2.807
24	0.256	0.531	0.857	1.318	1.711	2.064	2.492	2.797
25	0.256	0.531	0.856	1.316	1.708	2.060	2.485	2.787
26	0.256	0.531	0.856	1.315	1.706	2.056	2.479	2.779
27	0.256	0.531	0.855	1.314	1.703	2.052	2.473	2.771
28	0.256	0.530	0.855	1.313	1.701	2.048	2.467	2.763
29	0.256	0.530	0.854	1.311	1.699	2.045	2.462	2.756
30	0.256	0.530	0.854	1.310	1.697	2.042	2.457	2.75
40	0.255	0.529	0.851	1.303	1.684	2.021	2.423	2.704
60	0.254	0.527	0.848	1.296	1.671	2.000	2.39	2.66
120	0.254	0.526	0.845	1.289	1.658	1.980	2.358	2.617
∞	0.253	0.524	0.842	1.282	1.645	1.960	2.326	2.576

Question 4.DS.8: In an effort to overcome the noise problem presented by his ......

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