Probability and Statistics Lab 9

Info

IMG-20250326171708902.pdf

Questions & Answers

Question 1

A machine is set to produce metal plates of thickness 1.5 cms with standard deviation of 0.2 cm. A sample of 100 plates produced by the machine gave an average thickness of 1.52 cms. Is the machine fulfilling the purpose? Write a R program for above problem.

$\bar{X}=1.52cms$ $\mu =1.5cms$ $\sigma =0.2cm$ $n=100$ $\text{Level of Significance}=\alpha =0.05$

Confidence Interval

The confidence interval is the range of values within which the true value of the parameter is expected to lie with a certain level of confidence. The confidence interval is denoted by $1-\alpha$ where $\alpha$ is the level of significance. $(\bar{X}-Z_{\alpha }\frac{\sigma }{\sqrt{n}},\bar{X}+Z_{\alpha }\frac{\sigma }{\sqrt{n}})$

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# Sample data
mu = 1.5          # Population mean
x_bar = 1.52      # Sample mean
sigma = 0.2        # Standard deviation
n = 100            # Sample size
alpha = 0.05       # Level of significance
 
# Z-test statistic
z_stat = (x_bar - mu) / (sigma / sqrt(n))
print(paste("Z-statistic:", z_stat))
 
# P-value for two-tailed test
p_value = 2 * pnorm(-abs(z_stat))
print(paste("P-value:", p_value))
 
# Decision
if (p_value < alpha) {
  print("Reject Null Hypothesis: The machine is NOT working to the standards set.")
} else {
  print("Fail to Reject Null Hypothesis: The machine is working to the Standards set.")
}
 

Question 2

The average marks scored by 32 boys are 72 with SD of 8, while that for 36 girls is 70 with SD of 6. Test at 1% LOS whether the boys perform equal as girls.

	Boys	Girls
$\bar{X}$	72	70
$N$	32	36
$\sigma$	8	6

$\alpha =0.01$

Hypothesis $H_0:{\mu }_1={\mu }_2\to$ Boys and girls perform equally $H_1:{\mu }_1\ne {\mu }_2\to$ Boys and girls do not perform equally

$T=\frac{\left({\bar{X}}_1-{\bar{X}}_2\right)}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}$ From Hypothesis and Probability and Statistics

# Sample data
x1 <- 72        # Mean of boys
s1 <- 8         # SD of boys
n1 <- 32        # Sample size of boys
 
x2 <- 70        # Mean of girls
s2 <- 6         # SD of girls
n2 <- 36        # Sample size of girls
 
alpha <- 0.01   # Level of significance
 
# t-test statistic
t_stat <- (x1 - x2) / sqrt((s1^2 / n1) + (s2^2 / n2))
print(paste("t-statistic:", t_stat))
 
# Degrees of freedom (Welch-Satterthwaite)
df <- ((s1^2 / n1 + s2^2 / n2)^2) /
      ((s1^2 / n1)^2 / (n1 - 1) + (s2^2 / n2)^2 / (n2 - 1))
print(paste("Degrees of freedom:", df))
 
# P-value (two-tailed test)
p_value <- 2 * pt(-abs(t_stat), df)
print(paste("P-value:", p_value))
 
# Decision
if (p_value < alpha) {
  print("Reject Null Hypothesis: Boys and girls perform differently.")
} else {
  print("Fail to Reject Null Hypothesis: Boys and girls perform equally.")
}
 

Testing of Hypothesis Sample

References

Information

• date: 2025.03.25
• time: 12:27