4 An engineer wants to compane the tensle sarengths of stel bars that are producedusing a conventional method and an experimental method. (The tensle strength ofametal is a measure of its ability to resist tearing when pulled lengtheise. To do sa,the engineer randomly selects steel bars that are manufactured usng each methodand records the following tensle strengths (n newtons per square milimeter. Ata-0.10, can the engineer dam that the esperimental method produces steel withgreater mean tensle strength? Should the engineer recommend using theexperimental method? Assume the population variances are net equal(Experimental Method:395 389 421 394 407 411 389 412 422416 412 408 400 386 411 405 389Conventional Method:362 352 380 382 413 384 400378 419 379 384 388 372 363

Answered: 4 An engineer wants to compane the…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

4 An engineer wants to compane the tensle sarengths of stel bars that are produced
using a conventional method and an experimental method. (The tensle strength ofa
metal is a measure of its ability to resist tearing when pulled lengtheise. To do sa,
the engineer randomly selects steel bars that are manufactured usng each method
and records the following tensle strengths (n newtons per square milimeter. Ata
-0.10, can the engineer dam that the esperimental method produces steel with
greater mean tensle strength? Should the engineer recommend using the
experimental method? Assume the population variances are net equal(
Experimental Method:
395 389 421 394 407 411 389 412 422
416 412 408 400 386 411 405 389
Conventional Method:
362 352 380 382 413 384 400
378 419 379 384 388 372 363

Expert Solution

Step 1

Given that:

Experimental method:

395, 389, 421, 394, 407, 411, 389, 402, 422, 416, 402, 408, 400, 386, 411, 405, 389

Conventional method:

362, 352, 380, 382, 413, 384, 400, 378, 419, 379, 384, 388, 372, 383

Step 2

For experimental method:

Evaluate $Σ x a n d Σ x^{2}$ by using the following table:

x	$x^{2}$
395	156025
389	151321
421	177241
394	155236
407	165649
411	168921
389	151321
402	161604
422	178084
416	173056
402	161604
408	166464
400	160000
386	148996
411	168921
405	164025
389	151321
$Σ x = 6847$	$Σ x^{2} = 2759789$

Compute the sample mean by using the formula:

$\begin{array}{rcl} {\bar{x}}_{1} & = & \frac{Σ x}{n} \\ = & \frac{6847}{17} \\ = & 402.76 \end{array}$

Compute the sample standard deviation by using the formula:

$\begin{array}{rcl} s_{1} & = & \sqrt{\frac{Σ x^{2} - \frac{{(Σ x)}^{2}}{n}}{n_{1} - 1}} \\ = & \sqrt{\frac{2759789 - \frac{{(6847)}^{2}}{17}}{17 - 1}} \\ = & 11.34 \end{array}$

For Conventional method:

Evaluate $Σ x a n d Σ x^{2}$ by using the following table:

x	x2
362	131044
352	123904
380	144400
382	145924
413	170569
384	147456
400	160000
378	142884
419	175561
379	143641
384	147456
388	150544
372	138384
383	146689
$Σ x = 5376$	$Σ x^{2} = 2068456$

Compute the sample mean by using the formula:

$\begin{array}{rcl} {\bar{x}}_{2} & = & \frac{Σ x}{n} \\ = & \frac{5376}{14} \\ = & 384 \end{array}$

Compute the sample standard deviation by using the formula:

$\begin{array}{rcl} s_{2} & = & \sqrt{\frac{Σ x^{2} - \frac{{(Σ x)}^{2}}{n_{2}}}{n_{2} - 1}} \\ = & \sqrt{\frac{2068456 - \frac{{(5376)}^{2}}{14}}{14 - 1}} \\ = & 17.69 \end{array}$

Consider $μ_{1} a n d μ_{2}$ to be the population means of experimental method group and conventional method group respectively.

State the null and alternative hypotheses:

$\begin{array}{rcl} H_{0} : μ_{1} & = & μ_{2} \\ H_{1} : μ_{1} & > & μ_{2} \end{array}$

This is right tailed test.

Since the population standard deviation is not present so use t-test instead of z-test.

Compute the test statistic :

$\begin{array}{rcl} t & = & \frac{({\bar{x}}_{1} - {\bar{x}}_{2}) - (μ_{1} - μ_{2})}{\sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}} \\ = & \frac{(402.76 - 384) - (0)}{\sqrt{\frac{11 . 34^{2}}{17} + \frac{17 . 69^{2}}{14}}} \\ = & 3.429 \end{array}$