Respuesta :
There is sufficient data to prove that the heart rate was less than 120 beats per minute (P-value = 0.002).
For the population mean weight control heart rate of 20-year-olds, the 95% confidence interval is (98.07, 115.93).
Step-by-step explanation:
This is a test of the population mean's theory.
The assertion is that a heart rate of less than 120 bpm exists.
The alternative and null hypothesis are then:
h0 : μ = 120
ha : μ < 120
The level of significance is 0.05.
The sample has a n=100 size.
M=107 is the sample mean.
Due to the unknown population standard deviation, we estimate it using the sample standard deviation, which has a value of s=45.
Using the following formula, the estimated standard error of the mean is calculated:
Sm = [tex]\frac{s}{\sqrt{n} }[/tex]= [tex]\frac{45}{\sqrt{100} }[/tex] = 4.5
The t-statistic can then be calculated as follows:
T = [tex]\frac{m-μ}{\frac{s}{\sqrt{n} } }[/tex] = 107-120/4.5 = -13/4.5 -2.88
The t-statistic can then be calculated as follows:
df= n-1 = 100-1 = 99
These sample size's degrees of freedom are:
Given that this test has 99 degrees of freedom, a left-tailed design, and a t value of -2.889, the P-value is determined as follows (using a t-table)
p-value = p(t<-2.88)= 0.002
The effect is significant because the P-value (0.002) is less than the 0.05 level of significance.
The naive theory is disproved.
There is sufficient proof to back up the assertion that the heart rate was less than 120 bpm.
A 95% confidence interval for the mean must be calculated.
Since the population standard deviation is unknown, we must infer it from the sample standard deviation in order to get the critical number using a t-students distribution.
M=107 is the sample mean.
N=100 is the sample size.
When σ is unknown, an estimation of σM is made by dividing s by the square root of N:
Sm =[tex]\frac{s}{\sqrt{N} }[/tex] =[tex]\frac{45}{\sqrt{100} }[/tex] =4.5
The t-value for a 95% confidence interval is t=1.984.
The margin of error (MOE) can be calculated as:
MOE = t*Sm = 1.984 * 4.5 = 8.92
Then, the lower and upper bounds of the confidence interval are:
LL = M - t*Sm = 107- 8.92 = 98.07
UL = M +t *Sm = 107 + 8.92 = 115.93
The 95% confidence interval for the population mean weight control heart rate of 20-year-olds is (98.07, 115.93).
To learn more about T-statistic:
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