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3 years ago

11

Mathematics

1 answer:

enot [183]3 years ago

6 0

Step-by-step explanation:

Since, k is negative, Hence, coefficient of x will always be positive.
Equating (4 – 6k)x² + (8 – 2k)x + 4=0 with ax² + bx + c =0, we find: a = (4-6k), b= (8-2k) & c = 4
f(x) = 0 has no real roots.

$\therefore \: {b}^{2} - 4ac < 0 \\ \\ \therefore \: {(8 - 2k)}^{2} - 4(4 - 6k) \times 4 < 0 \\ \\ \therefore \: 64 + 4 {k}^{2} - 32k - 64 + 96k< 0 \\ \\ \therefore \: 64 + 4 {k}^{2} - 32k - 64 \\ \\\therefore \: 4 {k}^{2} + 64k < 0 \\ \\ \therefore \: 4 k(k + 16) < 0 \\ \\ \therefore \: k(k + 16) < 0 \\ \\ \therefore \: k < 0 \: \: or \: k + 16< 0 \\ \\ \therefore \: k < 0 \: \: or \: k < - 16\\ \\ \therefore \: k < - 16 \\ \\ \therefore \: k \in \{ - \infty, \: \: - 16 \}$

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90% confidence interval: (2186.53;2881.47)

95% confidence interval: (2118.73;2949.27)

99% confidence interval: (1987.37;3080.63)

For the last part is not the best way say : "This interval describes the price of 95% of the rents of all the unfurnished one-bedroom apartments in the Boston area."

The best interpretation is this one: "We are 95% confident that the actual mean for the rents of unfurnished one-bedroom apartments in the Boston area is between (2118.73;2949.27)"

Step-by-step explanation:

1) Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

$\bar X=2534$ represent the sample mean

$\mu$ population mean (variable of interest)

$\sigma=670$ represent the population standard deviation

n=10 represent the sample size

90% confidence interval

The confidence interval for the mean is given by the following formula:

$\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$ (1)

Since the Confidence is 0.90 or 90%, the value of $\alpha=0.1$ and $\alpha/2 =0.05$ , and we can use excel, a calculator or a tabel to find the critical value. The excel command would be: "=-NORM.INV(0.05,0,1)".And we see that $z_{\alpha/2}=1.64$

Now we have everything in order to replace into formula (1):

$2534-1.64\frac{670}{\sqrt{10}}=2186.53$

$2534+1.64\frac{670}{\sqrt{10}}=2881.47$

So on this case the 90% confidence interval would be given by (2186.53;2881.47)

95% confidence interval

The confidence interval for the mean is given by the following formula:

$\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$ (1)

Since the Confidence is 0.95 or 95%, the value of $\alpha=0.05$ and $\alpha/2 =0.025$ , and we can use excel, a calculator or a tabel to find the critical value. The excel command would be: "=-NORM.INV(0.025,0,1)".And we see that $z_{\alpha/2}=1.96$

Now we have everything in order to replace into formula (1):

$2534-1.96\frac{670}{\sqrt{10}}=2118.73$

$2534+1.96\frac{670}{\sqrt{10}}=2949.27$

So on this case the 95% confidence interval would be given by (2118.73;2949.27)

99% confidence interval

The confidence interval for the mean is given by the following formula:

$\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$ (1)

Since the Confidence is 0.99 or 99%, the value of $\alpha=0.01$ and $\alpha/2 =0.005$ , and we can use excel, a calculator or a tabel to find the critical value. The excel command would be: "=-NORM.INV(0.005,0,1)".And we see that $z_{\alpha/2}=2.58$

Now we have everything in order to replace into formula (1):

$2534-2.58\frac{670}{\sqrt{10}}=1987.37$

$2534+2.58\frac{670}{\sqrt{10}}=3080.63$

So on this case the 99% confidence interval would be given by (1987.37;3080.63)

For the last part is not the best way say : "This interval describes the price of 95% of the rents of all the unfurnished one-bedroom apartments in the Boston area."

The best interpretation is this one: "We are 95% confident that the actual mean for the rents of unfurnished one-bedroom apartments in the Boston area is between (2118.73;2949.27)"

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3 years ago