Answer :
To solve the problem of finding the probability [tex]\( P(z < 0.7) \)[/tex] for a standard normal distribution, follow these steps:
1. Understand the Standard Normal Distribution:
The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. The cumulative distribution function (CDF) for the standard normal distribution, denoted as [tex]\( \Phi(z) \)[/tex], gives the probability that a standard normal random variable is less than or equal to a given value [tex]\( z \)[/tex].
2. Identify the Z-value:
Here, the given Z-value is [tex]\( 0.7 \)[/tex].
3. Locate the CDF value for Z = 0.7:
For the Z-value of [tex]\( 0.7 \)[/tex], we want to find the cumulative probability [tex]\( \Phi(0.7) \)[/tex]. This represents the area under the standard normal curve to the left of [tex]\( z = 0.7 \)[/tex].
4. Calculate the CDF value:
By looking up standard normal distribution tables or using statistical software, we can determine [tex]\( \Phi(0.7) \)[/tex]. The value of the cumulative distribution function at [tex]\( z = 0.7 \)[/tex] is found to be [tex]\( 0.758 \)[/tex].
5. Express the Result:
Thus, the probability that [tex]\( z \)[/tex] is less than [tex]\( 0.7 \)[/tex] in a standard normal distribution is:
[tex]\[ P(z < 0.7) = 0.758 \][/tex]
In conclusion, the probability [tex]\( P(z < 0.7) \)[/tex], expressed as a decimal rounded to 4 decimal places, is [tex]\( 0.7580 \)[/tex].
1. Understand the Standard Normal Distribution:
The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. The cumulative distribution function (CDF) for the standard normal distribution, denoted as [tex]\( \Phi(z) \)[/tex], gives the probability that a standard normal random variable is less than or equal to a given value [tex]\( z \)[/tex].
2. Identify the Z-value:
Here, the given Z-value is [tex]\( 0.7 \)[/tex].
3. Locate the CDF value for Z = 0.7:
For the Z-value of [tex]\( 0.7 \)[/tex], we want to find the cumulative probability [tex]\( \Phi(0.7) \)[/tex]. This represents the area under the standard normal curve to the left of [tex]\( z = 0.7 \)[/tex].
4. Calculate the CDF value:
By looking up standard normal distribution tables or using statistical software, we can determine [tex]\( \Phi(0.7) \)[/tex]. The value of the cumulative distribution function at [tex]\( z = 0.7 \)[/tex] is found to be [tex]\( 0.758 \)[/tex].
5. Express the Result:
Thus, the probability that [tex]\( z \)[/tex] is less than [tex]\( 0.7 \)[/tex] in a standard normal distribution is:
[tex]\[ P(z < 0.7) = 0.758 \][/tex]
In conclusion, the probability [tex]\( P(z < 0.7) \)[/tex], expressed as a decimal rounded to 4 decimal places, is [tex]\( 0.7580 \)[/tex].
