Question

Concerns about climate change and CO2 reduction have initiated the commercial production of blends of biodiesel (e.g., from renewable sources) and petrodiesel (from fossil fuel). Random samples of 35 blended fuels are tested in a lab to ascertain the bio/total carbon ratio. (a) If the true mean is 0.9480 with a standard deviation of 0.0060, within what interval will 95 percent of the sample means fall? (Round your answers to 4 decimal places.)

Answer 1

Answer: $(0.9460\ ,\ 0.9500)$

Step-by-step explanation:

Given : Population mean : $\mu=0.9480$

Standard deviation : $\sigma=0.0060$

Sample size : n=35

Significance level : $\alpha=1-0.95=0.05$

Critical value = $z_{\alpha/2}=\pm1.96$

The confidence interval for sample mean is given by :-

$\mu\pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}$

i.e. $0.9480\pm (1.96)\dfrac{0.0060}{\sqrt{35}}$

$\approx0.9480\pm0.0020=(0.9480-0.0020\ ,\ 0.9480+0.0020)\\\\=(0.9460\ ,\ 0.9500)$

Hence, the required interval=  $(0.9460\ ,\ 0.9500)$

Answer 2

The 95% confidence interval for a random sample of 35 blended fuels is (0.9460 ; 0.9500)

Given the Parameters :

• Population mean, μ = 0.9480

• Standard deviation, σ = 0.0060

• Sample size, n = 35

• Zcritical, Z* = 1 - 0.95 = 0.05 $= Z_{0.05} = ±1.96$

The confidence interval is defined thus :

• μ ± Z*(σ/√n)

Confidence interval = 0.9480 ± 1.96(0.0060/√35)

Confidence interval = 0.9480 ± 0.0020

• Lower boundary = 0.9480 - 0.00020 = 0.9460

• Upper boundary = 0.9480 + 0.0020 = 0.9500

Therefore, the confidence interval is (0.9460 ; 0.9500)

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