Question

Question 1: A mobile phone manufacturer claims that the batteries in the manufactured mobile phones are used for an average of 140 hours after being charged once. For this purpose, 17 telephone batteries were chosen randomly and it was determined that they could be used for an average of 136 hours and the standard deviation was 29 hours. According to this;

a-) Is the batteries lasting less than 140 minutes according to 1% significance level? Examine statistically.

b-) Determine the confidence interval of the batteries' standby time at 1% significance level.

Answer 1

Solution :

a) The null and alternative hypotheses are as follows:

i.e. The average lasting time of batteries is 140 hours.

i.e. The average lasting time of batteries is less than 140 hours.

To test the hypothesis we shall use one sample t-test. The test statistic is given as follows :

Where, x̄ is sample mean, s is sample standard deviation, μ is hypothesized value of population mean under H​​​​​​​​0 and n is sample size.

We have,  x̄ = 136 hours, s = 29 hours , μ = 140 hours and n = 17

The value of the test statistic is -0.5687.

Since, our test is left-tailed test, therefore we shall obtain left-tailed p-value for the test statistic. The left-tailed p-value is given as follows :

p-value = P(T < t)

p-value = P(T < -0.5687)

Using the R software we get, P(T < -0.5687) = 0.2887

Hence, p-value = 0.2887

The p-value is 0.2887.

We make decision rule as follows :

If p-value is less than the significance level, then we reject the null hypothesis (H​​​​​​​​​0) at given significance level.

If p-value is greater than the significance level, then we fail to reject the null hypothesis (H​​​​​​​​​​​​​​​​0) at given significance level.

We have, p-value = 0.2887

significance level = 1% = 0.01

(0.2887 > 0.01)

Since, p-value is greater than the significance level of 1%, therefore we shall be fail to reject the null hypothesis (H​​​​​​​​​​​​​​​​0) at 1% significance level.

Conclusion : At 1% significance level, there is not sufficient evidence to conclude that battery lasting time is less than 140 hours.

b) The confidence interval for population mean at 1% significance level (at 99% confidence level) is given as follows:

Where, x̄ is sample mean, s is sample standard deviation, n is sample size and t(0.01/2, n -1) is critical t-value at 1% significance level and n - 1 degrees of freedom.

We have, x̄ = 136 hours, s = 29 hours and n = 17

Using t-table we get, t(0.01/2, 17 - 1) = 2.9208

Hence, confidence interval of the batteries' standby time at 1% significance level is,

The confidence interval of the batteries' standby time at 1% significance level is (115.46 hours, 156.54 hours).

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