A report released in May 2005 by First Data Corp. indicated that 43 % of adults had received a "

step 1 Right, so we're looking at data security and fishing emails. This is from a 2005 report, but tha

A report released in May 2005 by First Data Corp. indicated...

Key Concepts

Sampling Distribution of a Proportion

This concept refers to the probability distribution of a sample proportion, which estimates the true proportion in a population. For a large enough sample, the sample proportion will vary around the true population proportion and its distribution can be used to infer probabilities related to the sample outcome.

Central Limit Theorem

The Central Limit Theorem states that the distribution of sample means or proportions will approximate a normal distribution as the sample size increases, regardless of the underlying population distribution. This theorem is fundamental when approximating the sampling distribution of a proportion, especially in large samples.

Normal Approximation to the Binomial Distribution

When a sample is large, the binomial distribution, which describes the number of successes in a fixed number of independent Bernoulli trials, can be approximated by a normal distribution. This simplifies the calculation of probabilities, such as the likelihood of obtaining a specific range of sample proportions.

Standard Error of a Proportion

The standard error measures the variability of a sample proportion around the true population proportion. It is calculated using the formula sqrt[p(1-p)/n], where p is the population proportion and n is the sample size. This value is crucial for assessing how much the sample proportion might fluctuate due to random sampling.

Z-Score Calculation

A z-score represents the number of standard errors a sample observation is away from the population mean. In the context of a sample proportion, it is used to standardize the difference between the observed proportion and the true proportion, allowing for the calculation of probabilities using the standard normal distribution.

Tail Probabilities and Unusual Events

This concept involves determining the probability that a sample proportion falls within a specific tail of the sampling distribution (either very low or very high). An event is considered unusual if it occurs with a very low probability, typically below a threshold such as 5%, which can indicate that such an outcome is unlikely under the assumed conditions.

Transcript

00:01 Right, so we're looking at data security and fishing emails.

00:05 This is from a 2005 report, but that's still going on today, and it's 2021.

00:11 So this is kind of a big deal still.

00:16 So let's see, we're told that 42 % of adults have received a fishing contact, which is some bogus email.

00:26 And a study, a simple random sample of 800 adults, is a...

00:33 And we want to know a few things the probability that no more than 40 % that received this kind of an email so in previous exercises you're asked to do to check if it's normal and it's going this this ends up being it i did the the check here i had the spreadsheet set up that we input in the p which is point four three the n which is 800 there and then you just substitute that in here and it ends up being greater than a equal to 10.

01:10 That's not part of the exercise, but in order to proceed, that has to be true.

01:15 So i just kind of just point that out here.

01:20 And then the size needs to be appropriate so that the sample size needs to be less than 5 % of the total population for all this stuff to work.

01:27 And 800 is well under the 5 % of all the adults in the us.

01:34 So we're good there.

01:35 All right, so let's get to it.

01:38 We have the probability that no more than 40 % have received this email.

01:43 So we want capital p that no more than 43%, so p hat is less than or equal to 40%.

01:57 Well, p is going to be a nut percentage.

02:00 It's 0 .4.

02:05 So we need to figure that out.

02:07 So let's do that.

02:07 So because it's normal, we know that the mean of the sample distribution is p, and the standard deviation is this square root of the quantity p times 1 minus p over n.

02:25 And i've done that here in the spreadsheet because spreadsheets are just nifty.

02:29 So there's what it is, the square root of p times 1 minus p all over n.

02:35 And there we go.

02:36 So we get this is our sigma.

02:37 So i'll write this out.

02:41 So we know that p, oops, that mu of p hat is .43 and sigma of p hat is you just substitute in the p and then here and you get .0175.

03:00 And then we need to figure out the z score, z which is the p hat value we're concerned with.

03:10 0 .40, minus 0 .43 all over.

03:17 .1175.

03:19 And yes, i have done that in the spreadsheet, but it's not set up correctly.

03:24 So i've got to do that...

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