Question 1

Could any one explain how to get the p-value in the second example?

Accepted Answer

Sure!

The p-value is the probability of a statistic at least as deviant as ours occurring under the assumption that the null hypothesis is true.

Under that assumption, and noting also that we are given that the population is normally distributed (or that we took a sample size of at least 30 [by the Central Limit Theorem]), we can treat the sampling distribution of the sample mean as a normal distribution.

So now, we can use the normal cumulative density function or a z-table to find this probability. (We could also use a t-table, but it is allowable to just use a z table since our sample size is larger than 30)

To use a z-table, we'll need to find the appropriate z-score first.

Since the answer to what we are asking comes from the sampling distribution of the sample mean, we would find the appropriate standard deviation to use by dividing the population standard deviation by the square root of the sample size (since the variance of the sampling distribution is the population variance divided by the sample size, and the standard deviation is the square root of the variance).

That would give us a standard deviation for the sampling distribution of the sample mean.

I say would, because unfortunately, we don’t always know the population standard deviation, and so (as it seems they did here, despite knowing the population standard deviation), we are using the sample standard deviation in its place to find an estimate of the standard deviation for the sampling distribution of the sample mean, which is also known as the standard error of the mean.

In our example, the standard error of the mean therefore has a value of 0.12 / 50^0.5, or approximately 0.01697.

Taking the difference between our sample mean and the population mean and dividing it by the standard error gives us our z-score (number of standard errors our sample mean is away from the population mean), which is approximately (7.36 - 7.4) / 0.01697 or -2.36.

Since the alternative hypothesis is not specific about the population mean being either greater than or less than the value in the null hypothesis, we have to consider both tails of the distribution, but by symmetry of the standard normal distribution, we can accomplish this by simply doubling the value we get from using our obtained z-score with a z-table.

The value given by a z-table using a z-score of -2.36 is 0.0091, which, when doubled, is 0.0182 or approximately 0.02.

This (or other videos before it in that section) might also help (it comes later in this unit): https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/v/calculating-p-value-from-t-statistic

:)

Question 2

I don't understand the p-value in example 1

Isn't the calculation: binomial(60,20) * 0.75^40 * 0.25^20 = 0.0383?
The problem states it is "0.068". Is this p-value wrong or did I make a mistake in my calculation?

Accepted Answer

p-value = P(p(x) >= 20/60 given that the actual proportion is 0.25)
So, You need to calculate:
binomial(60,20) * 0.75^40 * 0.25^20  (probability of  20 subject that identified the bottled water )
                                  +
binomial(60,21) * 0.75^39 * 0.25^21  (probability of  21 subject that identified the bottled water )
                                   +
binomial(60,22) * 0.75^38 * 0.25^22 
                                   +
binomial(60,23) * 0.75^37 * 0.25^23
                                   :
                                   :
binomial(60,60) * 0.75^0 * 0.25^60 (probability of all the 60 subjects identified the bottled water )

I guess !

Question 3

How do you decide what Significance level you should set??

Accepted Answer

A significance level of 0.05 (i.e. 5%) is commonly used, but sometimes other significance levels are used.

Note that the significance level is the probability of a Type 1 error (rejecting a true null hypothesis).  Everything else being equal, decreasing the significance level (probability of a Type 1 error) increases the probability of a Type 2 error (failing to reject a false null hypothesis), and vice versa.

So the statistician has to weigh the cost of a Type 1 error (rejecting a true null hypothesis) versus the cost of a Type 2 error (failing to reject a false null hypothesis) in the real-world situation.  If the statistician is especially concerned about the cost of a Type 1 error, then he/she will use a significance level that is less than 0.05. However, if instead the statistician is especially concerned about the cost of a Type 2 error, then he/she will use a significance level that is greater than 0.05.

Question 4

In the first problem, is 0.068 the correct p-value? Assuming that the null hypothesis is true, and p = 0.25, the sampling distribution of sample proportion with n = 60 should be approximately normal, with a mean = p = 0.25 and standard deviation of √((p·(1-p))/n) ≈ 0.056. So a sample with p-hat = 0.3 should only have a z-score ≈ 0.89, and there should be ≈ 0.187 probability of getting a sample with p-hat ≥ 0.3. Or am i missing something?

Accepted Answer

You generally had the right idea for calculating the p-value. Note that the p-hat value is not 0.3, but rather 20/60 = 1/3 = 0.3333... (perhaps you did not consider the bar on top of the decimal digit 3). So the z-score is about 1.49 instead of 0.89. The probability of equaling or exceeding a z-score of 1.49 is about 0.068.

Have a blessed, wonderful day!

Have a blessed, wonderful day!

Question 5

Please please please show us how those P Values come about. We are very troubled.

Accepted Answer

the p values are generally given whenever such problems are asked, i think calculating p values is a completely unrelated concept here so it is not taught

Question 6

ur... are we going to be told how to calculate this P-value?

I'm confused on what it actually is...

Accepted Answer

Yes, there are lessons on how to calculate the p-value, which is the probability that the assumed population parameter is true, based on a sample statistic of that parameter.

AP®︎/College Statistics

Course: AP®︎/College Statistics > Unit 10

Using P-values to make conclusions

Example 1

Example 2

Ethics and the significance level $α$ ‍

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