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3 years ago

Express 0.484848..... in p/q form

Mathematics

1 answer:

cupoosta [38]3 years ago

7 0

<h3>Answer: 16/33</h3>

It's in p/q form where p = 16 and q = 33.

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Work Shown:

x = 0.484848.....

100x = 48.4848.....

I multiplied both sides by 100 to move the decimal over 2 spots. Both decimal values for x and 100x have an infinite string of "48"s repeated after the decimal point. When we subtract, those infinite strings will cancel out

100x - x = 99x

48.4848..... - 0.484848..... = 48

So after subtracting straight down, we have the new equation 99x = 48 which solves to x = 48/99

Divide both parts by the GCF 3 to fully reduce

48/3 = 16

99/3 = 33

Therefore, x = 48/99 = 16/33 = 0.484848...

I recommend using a calculator to confirm that 16/33 = 0.484848...

Side note: your calculator may round the last digit, but this is of course rounding error

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$0.0497 - 1.96\sqrt{\frac{0.0497(1-0.0497)}{322}}=0.026$

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The 90% confidence interval would be given by (0.026;0.073)

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".

The population proportion have the following distribution

$\hat p \sim N(p,\sqrt{\frac{p(1-p)}{n}})$

Solution to the problem

The estimated proportion for this case is:

$\hat p =\frac{322-306}{322}= 0.0497$

Represent the proportion of defectives for this case

In order to find the critical value we need to take in count that we are finding the interval for a proportion, so on this case we need to use the z distribution. Since our interval is at 90% of confidence, our significance level would be given by $\alpha=1-0.90=0.1$ and $\alpha/2 =0.05$ . And the critical value would be given by: