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Class 12 Applied Maths Chapter 10 Solutions
Inferential Statistics
EXERCISE- 10.3
Q.1 Consider the following hypothesis test:
Η0 : μ ≥ 80
Ηa : μ < 80
A sample of 100 is used and the population standard deviation is 12. Compute the p-value and state your conclusion for each of the following sample results, use a = 0.01:
(i) x̄ = 77
(ii) x̄ = 81
Ans. σ = 12, n = 100, α = 0.01
We will compute the Z-score and corresponding p-value and compare it with α = 0.01
(i) x̄ = 77
Z = 
= 
= 
= 
Therefore, Z < 0
P value = Area to left of Z table
= 0.0062
P value < α
Therefore, Reject H0 .
(ii) x̄ = 81
Z =
= 
= 
= = 0.83
Therefore, Z > 0
P value = Right of Z table
= 1 – 0.7967
= 0.2033
P value > α
Therefore, Do not Reject H0
Q.2 Consider the following hypothesis test:
H0: μ = 22
Ηa : μ ≠ 22
A sample of 75 is used and the population standard deviation is 10. Compute the p-value and state your conclusion for each of the following sample results, use α = 0.01:
(i) x̄ = 23
(ii) x̄ = 25.1
Ans. σ = 10, n = 75, α = 0.01
(i) x̄ = 23
Z =
= 
= 
= 
= 0.866 = 0.87
Therefore, Z < 0
P value = 2 (Area to right of Z table)
= 2 (1 – 0.8078)
= 2 x 0.1922
= 0.3844
P value > α
Therefore, Do not Reject H0 .
(ii) x̄ = 25.1
Z =
= 
= 
= 
Therefore, Z > 0
P value = 2 (Area to Right of Z table)
= 2 (1 – 0.9965)
= 0.0070
P value < α
Therefore, Reject H0
Q.3 Consider the following hypothesis test:
H0 : μ ≤ 50
Ha : μ > 50
A sample of 60 is used and the population standard deviation is 8. Use the critical value approach to state your conclusion for each of the following sample results, use α = 0.05
(i) x̄ = 52.5
(ii) x̄ = 51
Ans. n = 60, σ = 8, α = 0.05, μ0 = 50
(i) x̄ = 52.5
Z =
= 
= 
= 2.42
Compare with critical value:
Zα = Z0.05 = 1.645
Since, Z = 2.42 > 1.645
Therefore, Reject H0 .
(ii) x̄ = 51
Z =
= 
= 
= 0.97
Compare with critical value:
Zα = Z0.05 = 1.645
Since, Z = 0.97 < 1.645
= Z < Zα
Therefore, Do not Reject H0 .
Q.4 Consider the following hypothesis test:
H0 : μ ≥ 1056
Ha : μ < 1056
A sample of 400 provided a sample mean of 910. The population standard deviation is 1600.
(i) Compute the value of the test statistic.
(ii) What is the p-value?
(iii) At α = 0.05, what is your conclusion?
(iv) What is the rejection rule using the critical value? What is your conclusion?
σ = 1600, n = 400, x̄ = 910
(i) Z =
= 
= 
= -1.83
(ii) Z < 0
P value = 0.0336
(iii) α = 0.05
P value < α
Therefore, Reject H0 .
(iv) Using Critical Value:
Z0.05 = −1.645
Since Z = −1.825 < −1.645
Therefore, Reject H0 .
Q.5 Consider the following hypothesis test:
H0 : μ = 8
Ηa : μ ≠ 8
A sample of 120 provided a sample mean of 8.4. The population standard deviation is 3.2.
(i) Compute the value of the test statistic.
(ii) What is the p-value?
(iii) At α = 0.05, what is your conclusion?
(iv) Compute 95% confidence interval for the population mean. Does it support your conclusion?
Ans. x̄ = 8.4, σ = 3.2, n = 120
(i) Z =
= 
= 
= 
= 1.37
(ii) Z > 0
P value = 2 (Right of Z)
= 2 (1 – 0.9147)
= 2 x 0.0853
= 0.1706
(iii) α = 0.05
P value > α
Therefore, Do not Reject H0 .
(iv) Margin of Error = Zα/2 x 
= Z0.025 x 
= 1.960 x 
= 
Now, Confidence Interval :
µ = x̄ – E , µ = x̄ + E
= 8.4 – 0.57 , = 8.4 + 0.57
= 7.83 , = 8.97
(7.83 , 8.97)
Q.6 Consider the following hypothesis test:
H0 : p = 0.20
Ηa : p ≠ 0.20
A sample of 400 provided a sample proportion p̄ = 0.175
(i) Compute the value of the test statistic.
(ii) What is the p-value?
(iv) What is rejection rule using critical value? What is your conclusion?
Ans. p̄ = 0.175 , n = 400
(i) Z = 
= 
= 
= -1.25
(ii) Z < 0
P value = 2(left to Z)
= 2 (0.1056)
= 0.2112
(iii) α= 0.05
P value > α
Therefore, Do not Reject H0
(iv) Using Critical Value:
Z ≤ -Zα/2 Or, Z ≥ -Zα/2
Zα/2 = 1.960
Z < Zα/2
Therefore, Do not reject H0
Q.7 Consider the following hypothesis test:
H0 : p ≤ 0.24
Ha : p > 0.24
A sample of 300 provided a sample proportion of 0.31
(i) Compute the value of the test statistic.
(ii) What is the p-value?
(iii) At α = 0.05, what is your conclusion?
(iv) What is rejection rule using the critical value? What is your conclusion?
Ans. p̄ = 0.31 , n = 300
(i) Z =
= 
= 
= 2.84
(ii) Z > 0
P value = Area to the right of Z
= 1 – 0.9977
= 0.0023
(iii) α = 0.05
P value < α
Therefore, Reject H0
(iv) Using Critical Value:
Z ≥ Zα [Reject H0]
Zα = 1.645
Z > Zα
Therefore, Reject H0
Q.8 In India salaried individuals filling income tax returns received an average refund of Rs.1056. Consider the population of last minute income tax payers who file their tax return during the last five days of the income tax period.
(i) A researcher suggests that a reason individuals wait until last five days is that on average these individuals receive lower refunds than do early fillers.
Develop appropriate hypothesis such that rejection of H0 will support the researcher’s point of view.
(ii) For a sample of 400 individuals who filled a tax return in last five days, the sample mean refund was Rs.910. Based on prior experience a population standard deviation of σ = Rs.1600 may be assumed, what is the p-value?
(iii) At α = 0.05, what is your conclusion?
(iv) Repeat the preceding hypothesis test using the critical value approach.
Ans. Population mean refund: μ0 = 1056
Sample size: n = 400
Sample mean: x̄ = 910
Population standard deviation: σ = 1600
Significance level: α = 0.05
(i) H0 : μ ≥ 1056
Ha : μ < 1056
This is a left-tailed test.
(ii) We use the Z-test formula:
Z =
=
=
= -1.83
Z < 0
P value = 0.0336
(iii) α = 0.05
P value < α
Therefore, Reject H0 .
(iv) Using Critical Value:
Z0.05 = −1.645
Since Z = −1.825 < −1.645
Therefore, Reject H0 .
Q.9 ABC Infra an Indian real estate research firm, tracks the cost of apartment rentals in Mumbai. In mid 2016, the mean apartment rental rate was Rs.62650 per month in Mumbai. Assume that based on quarterly surveys, a population standard duration σ = Rs.15750 is reasonable. In a current study of apartment rental rates, a sample of 180 apartments in Mumbai has the mean rental rate Rs.61250 per month. Do the sample date enable ABC Infra to conclude that the population mean apartment rental rate now exceeds the level reported in 2016 ?
(i) State null and alternative hypothesis.
(ii) What is the p-value?
(iii) At α = 0.01, what is your conclusion?
Ans. Historical Mean (2016): μ0 = 62650
Sample size: n = 180
Sample mean: x̄ = 61250
Population standard deviation: σ = 15750
Significance level: α = 0.01
(i) H0 : μ ≤ 62650
Ha : μ > 62650
This is a left-tailed test.
(ii) We use the Z-test formula:
Z =
= 
=
= -1.19
Z < 0
P value = 0.1170
(iii) α = 0.01
P value > α
0.1170 > 0.01
Therefore, Do not Reject H0 .
Q.10 The average annual total return for SBI global advantage mutual fund from 1999 to 2003 was 4.1%. A researcher would like to conduct a hypothesis test to see whether the returns for PNB growth mutual funds over the same period are significantly different from the average for SBI global advantage mutual funds.
(i) Formulate the hypothesis that can be used to determine whether the mean annual return for PNB growth funds differ from the mean for SBI global advantage funds.
(ii) A sample of 40 PNB growth funds provides a mean return of x̄ = 3.4%. Assume the population standard deviation for PNB growth funds is known from previous studies to be σ = 2%. Use the sample results to compute the test statistic and p-value for the hypothesis test.
(iii) At a = 0.05, what is your conclusion?
Ans. Mean annual return for SBI Global Advantage Fund: μ0 = 4
Sample size of PNB Growth Funds: n = 40
Sample mean of PNB: x̄ = 3.4
Population standard deviation for PNB: σ = 2
Significance level: α = 0.05
(i) H0 : µ = 4.1
Ηa : µ ≠ 4.1
(ii) Z =
= 
= -2.21
Z < 0
P value = 0.0136
(iii) α = 0.05
P value < α
0.0136 < 0.05
Therefore, Reject H0 .
FAQ’s related to Class 12 Applied Maths Chapter 10 on Inferential Statistics:
Q1. What is Inferential Statistics?
Ans. Inferential Statistics is the branch of statistics that allows us to make predictions or inferences about a population based on a sample. It involves estimating parameters, testing hypotheses, and drawing conclusions using data analysis.
Q2. How is Inferential Statistics different from Descriptive Statistics?
Ans. (i) Descriptive Statistics deals with summarizing and presenting data (like mean, median, mode).
(ii) Inferential Statistics goes further to draw conclusions about the larger population from a small sample using tools like confidence intervals and hypothesis testing.
Q.3 What are the main concepts covered in this chapter?
Ans. (i) Population and Sample
(ii) Parameters and Statistics
(iii) Sampling methods
(iv) Types of errors (Type I and Type II)
(v) Confidence Interval
(vi) Hypothesis Testing
(vii) Level of Significance
Q.4 What is a population and sample in statistics?
Ans. (i) Population is the entire group you want to study (e.g., all students in India).
(ii) Sample is a subset taken from the population for analysis (e.g., 500 students selected randomly).
These are a few Frequently Asked Questions relating to Class 12 Applied Maths Chapter 10
In Class 12 Applied Maths chapter 10, you will explore fascinating topics that form the backbone of practical problem-solving techniques. Through clear explanations, illustrative examples, and step-by-step solutions, you’ll grasp complex concepts effortlessly. Whether you’re preparing for exams or simply eager to deepen your mathematical understanding, Class 12 Applied Maths Chapter 10 promises an enriching learning experience that will set you on the path to success. Class 12 Applied Maths Chapter 10, we delve deep into advanced mathematical concepts that are crucial for understanding.