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Anastasy [175]
3 years ago
15

The the length of a rectangle is 9 centimeters less than four timesfour times its width. its area is 28 square centimeters. find

the dimensions of the rectangle.
Mathematics
1 answer:
wel3 years ago
4 0

Let w = width

then

5w-3 = length

w(5w-3) = 14

5w^2-3w = 14

5w^2-3w-14=0

(5w+7)(w-2) = 0

w = {-7/5, 2}

throw out the negative solution leaving

w = 2 centimeters

length:

5w-3 =5(2)-3 =10-3 = 7 centimeters

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The diet which is developed by the nutritionist is effective at the 0.05 level of significance. Claim is agreed.

<h3>When do we use two-sample t-test?</h3>

The two-sample t-test is used to determine if two population means are equal.

A nutritionist has developed a diet that she claims will help people lose weight. In this,

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Here are the original weights, in pounds, with the weight after 6 months in parentheses.

  • Before 192 212 171 215 180 207 165 168 190 184 200 196
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The mean of the weights before 6 moths is,

\overline X_1=\dfrac{192+ 212 +171 +215 +180 +207 +165 +168 +190 +184 +200 +196 }{12}\\\overline X_1=190

The mean of the weights after 6 months is,

\overline X_2=\dfrac{ 183 +196  +174 +211 +160 +191   +162 +175 +190  +179  +189 +195  }{12}\\\overline X_1=183.75

Standard deviation of both the data is 16.9 and 14.7.

1. Null and Alternative Hypotheses.

The following null and alternative hypotheses need to be tested:

\begin{array}{ccl} H_0: \mu_1 & = & \mu_2 \\\\ \\\\ H_a: \mu_1 & > & \mu_2 \end{array}

This corresponds to a right-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

  • (2) Rejection Region

Based on the information provided, the significance level is α=0.05 and the degrees of freedom are df = 22. In fact, the degrees of freedom are computed as follows, assuming that the population variances are equal:

df_{Total} = df_1 + df_2 = 11 + 11 = 22

Hence, it is found that the critical value for this right-tailed test is

t_c=1.717, for α=0.05 and df=22

The rejection region for this right-tailed test is,

R = \{t: t > 1.717\}

(3) Test Statistics

Since it is assumed that the population variances are equal, the t-statistic is computed as follows:

t = \displaystyle \frac{\bar X_1 - \bar X_2}{\sqrt{ \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}(\frac{1}{n_1}+\frac{1}{n_2}) } }

\displaystyle \frac{ 190 - 183.75}{\sqrt{ \frac{(12-1)16.9^2 + (12-1)14.7^2}{ 12+12-2}(\frac{1}{ 12}+\frac{1}{ 12}) } } = 0.967

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Since it is observed that t=0.967≤tc=1.717 it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=0.1721 and since p=0.1721≥0.05p = 0.1721  it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis H₀ is not rejected. Therefore, there is not enough evidence to claim that the population mean μ1​ is greater than μ2​, at the α=0.05 significance level.

Confidence Interval

The 95% confidence interval is −7.16<μ<19.66

Thus, the diet which is developed by the nutritionist is effective at the 0.05 level of significance. Claim is agreed.

Learn more about the two-sample t-test here;

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