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4 years ago

Help super important

1 answer:

8 0

Group A:
1 dot at 9:1 x 9 = 9
7 dots at 10: 7 x 10 = 70
8 dots at 15: 8 x 15 = 120
8 dots at 17: 8 x 17 = 136
6 dots at 19: 6 x 19 = 114
Mean: (9+70+120+136+114) / (1+7+8+8+6) = 449/30 = 14.97

Group B:
6 dots at 10: 6 x 10 = 60
5 dots at 14: 5 x 14 = 70
6 dots at 18: 6 x 18 = 108
5 dots at 25: 5 x 25 = 125
4 dots at 28: 4 x 28 = 112
4 dots at 29: 4 x 29 = 116
mean: (60+70+108+125+112+116) / (6+5+6+5+4+4) = 591 / 30 = 19.70

Group A is the group that has the lower mean age because its mean is 14.97. I hope this helped ya :)

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Step-by-step explanation:

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Answer:

27.88% probability that a random sample of 5 has a mean between 677 and 693.

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$ , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$ .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem, we have that:

$\mu = 675, \sigma = 44, n = 5, s = \frac{44}{\sqrt{5}} = 19.6774$

The probability that a random sample of 5 has a mean between 677 and 693.

This is the pvalue of Z when X = 693 subtracted by the pvalue of Z when X = 677. So

X = 693

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{693 - 675}{19.6774}$

$Z = 0.91$

$Z = 0.91$ has a pvalue of 0.8186

X = 677

$Z = \frac{X - \mu}{s}$

$Z = \frac{677 - 675}{19.6774}$

$Z = 0.1$

$Z = 0.1$ has a pvalue of 0.5398

0.8186 - 0.5398 = 0.2788

27.88% probability that a random sample of 5 has a mean between 677 and 693.

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4 years ago

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3 years ago

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Answer:

a) $\hat \mu = \bar X = 145$

b) $145-1.64\frac{35}{\sqrt{25}}=133.52$

$145+1.64\frac{35}{\sqrt{25}}=156.48$

So on this case the 90% confidence interval would be given by (133.52;156.48)

Step-by-step explanation:

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".

$\bar X=145$ represent the sample mean

$\mu$ population mean (variable of interest)

$\sigma=35$ represent the population standard deviation

n=25 represent the sample size

a) For this case the best point of estimate for the population mean is the sample mean:

$\hat \mu = \bar X = 145$

b) Calculate the 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 table 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):

$145-1.64\frac{35}{\sqrt{25}}=133.52$

$145+1.64\frac{35}{\sqrt{25}}=156.48$

So on this case the 90% confidence interval would be given by (133.52;156.48)

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3 years ago