The following is the proportion of normal lung use for a random sample of 17 pat

Question

The following is the proportion of normal lung use for a random sample of 17 patients with acute tired lung: 0.24, 0.19, 0.40, 0.23, 0.30, 0.19, 0.24, 0.32, 0.28, 0.24 0.18, 0.22, 0.14, 0.30, 0.07, 0.12, 0.17 Suppose that the proportion of normal lung use for acute tired lung patients has an unknown mean of 1 and an unknown standard deviation . It is also known that tired lung proportions of normal lung use are normally distributed. a) Calculate the upper 10% percentile(Same as 90th percentile) of the t distribution with 16 degrees of freedom. b) Calate the upper 5% percentile(Same as 95th percentile) of the t distribution with 16 degrees of freedonm c) Calculate the upper 5% percentile(Same as 95th percentile) of the t distribution with 17 degrees of freedom. d) Calate the upper 5% percentile(Same as 95th percentile) of the standard normal distribution. e) Calculate the sample standard deviation for this data? f) Calculate the sample mean for this data g) Compute a 90% Confidence Interval for h)Compute a 90% Prediction Interval for a single future weight measurement.

Explanation / Answer

Solution:

Part a

We have to find upper 10% percentile or 90th percentile.

From given data, we have

Sample mean = Xbar = 0.225294118

Sample standard deviation = S = 0.080786574

Sample size = n = 17

Degrees of freedom = n – 1 = 16

Critical t value = 1.745883669

Required value = 90th percentile = Upper 10% percentile = X = Xbar + t*SD

Required value = 0.225294118 + 1.745883669*0.080786574

Required value = 0.366338078

Part b

We have to find upper 5% percentile or 95th percentile.

From given data, we have

Sample mean = Xbar = 0.225294118

Sample standard deviation = S = 0.080786574

Sample size = n = 17

Degrees of freedom = n – 1 = 16

Critical t value = 2.119905285

Required value = 95th percentile = Upper 5% percentile = X = Xbar + t*SD

Required value = 0.225294118 + 2.119905285*0.080786574

Required value = 0.396554003

Part c

We have to find upper 5% percentile or 95th percentile.

From given data, we have

Sample mean = Xbar = 0.225294118

Sample standard deviation = S = 0.080786574

Degrees of freedom = 17

Critical t value = 2.109815559

Required value = 95th percentile = Upper 5% percentile = X = Xbar + t*SD

Required value = 0.225294118 + 2.109815559*0.080786574

Required value = 0.395738889

Part d

We have to find upper 5% percentile or 95th percentile.

From given data, we have

Sample mean = Xbar = 0.225294118

Sample standard deviation = S = 0.080786574

Critical Z value = 1.644853627

Required value = 95th percentile = Upper 5% percentile = X = Xbar + Z*SD

Required value = 0.225294118 + 1.644853627*0.080786574

Required value = 0.358176207

[Note: All critical t values and z values are calculated by using z-table and t-table. We can also find these values by using simple excel commands =tinv(p,df) and =normsinv(p).]

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