According to the study, the mean salary of the workers is P 250 per day with standard deviation

According to the study, the mean salary of the workers is P...

According to the study, the mean salary of the workers is $\mathrm{P} 250$ per day with standard deviation of $\mathrm{P} 10 .$ Assuming that the distribution of salaries is normally distributed. a. What is the probability that the workers received a salary between $\mathrm{P} 245$ and $\mathrm{P} 275$ per day? b. What is the probability that the workers received a salary higher than or equal to $\mathrm{P} 285$ per day? c. What is the probability that the workers received a salary of lower than or equal to $\mathrm{P} 240$ per day? What is the probability that the workers received a salary between $\mathrm{P} 230$ and $\mathrm{P} 250$ per day?

According to the study, the mean salary of the workers is $\mathrm{P} 250$ per day with standard deviation of $\mathrm{P} 10 .$ Assuming that the distribution of salaries is normally distributed.
a. What is the probability that the workers received a salary between $\mathrm{P} 245$ and $\mathrm{P} 275$ per day?
b. What is the probability that the workers received a salary higher than or equal to $\mathrm{P} 285$ per day?
c. What is the probability that the workers received a salary of lower than or equal to $\mathrm{P} 240$ per day?
What is the probability that the workers received a salary between $\mathrm{P} 230$ and $\mathrm{P} 250$ per day?

Key Concepts

Probability Calculation for Intervals

Calculating the probability that a normally distributed variable falls within an interval involves determining the difference in the CDF values at the endpoints of the interval. This method is applied regardless of whether the interval represents a one-tailed or two-tailed probability scenario.

Standardization (Z-score Transformation)

Standardization involves converting a normally distributed random variable into a standard normal variable by subtracting the mean and dividing by the standard deviation. This process, known as computing the z-score, enables the application of standard normal distribution tables to calculate probabilities.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) for a normal distribution gives the probability that a random variable will be less than or equal to a particular value. It is exploited in calculating the area under the normal curve over specified intervals, which corresponds to the probability of the variable falling within that range.

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about its mean. It is defined by its mean (central tendency) and its standard deviation (measure of spread), and its bell-shaped curve indicates that values near the mean are more frequent in occurrence than values far from the mean.

Mean and Standard Deviation

The mean provides a measure of the central location of the data, while the standard deviation quantifies the amount of variation or dispersion from the mean. In a normal distribution, about 68% of the data falls within one standard deviation from the mean, which is a key indicator used in probability calculations.

Transcript

00:02 All right.

00:02 So we're given a scenario and we're assuming within this scenario that, in this case, the hourly wages follow a normal probability distribution.

00:12 And that's one of the things we do with statistics is we need to consider, does this follow a normal distribution? and if it does, then we can use the tools that we have.

00:27 And we're told it is normal distribution.

00:30 So we're good to go.

00:31 And we're told the mean is $20 .5.

00:36 And this is for hourly wages of maintenance crew, crew members for airlines.

00:44 So the mean is $20 .5.

00:48 And the standard deviation is $3 .5.

00:51 So we want to find what's the probability if we pick a crew member at random that they earn between $20 .5 and $24.

01:06 So we are going to find our z score and kind of put it to our distribution table in.

01:13 And let's go for it.

01:17 So the z score will then, it's going to be, let's see, the x in this case is 24.

01:30 So we do 24 minus the mu, which is 20 and a half, all over three and a half.

01:40 And in this case, and this is one of the reasons i have the spreadsheet opens so we can use that to do our calculations.

01:49 So the z score for this is going to be equal to by the way.

02:08 One.

02:08 Oh, wow.

02:09 Perfect.

02:10 You know what? i should have looked at this.

02:12 Holy cow.

02:12 My brain is not working well today.

02:17 24 is exactly $3 .5 a way.

02:20 So of course the standard deviation is one gee willickers all right look at that it's exactly one one stern deviation away um and we can actually figure this out pretty easily now than thinking about it it's funny when i did this initially i used all this spreadsheets but then as i put the picture to it my brain is clicking oh my gosh we could actually use our understanding of stern deviations and do it without much calculation at all.

02:59 So anyway, i did it.

03:00 So let's go through it.

03:02 So what you do is you find the, we use our form.

03:05 This would be the same thing as a table lookup, but i use this function called the normal dist.

03:09 So the x is 1.

03:12 The mean is 0 for the standard normal.

03:14 The standard deviation for the standard normal is 1.

03:17 Cumulative yes.

03:19 But we want to subtract off 0 .5 because that is this chunk right here.

03:24 This chunk right here to the left of the mean right here.

03:29 This is 0 .5.

03:30 And the z tables will give us everything up to that value.