# Hawaiians are known to have an active lifestyle, enjoying the fresh air and natural beauty the islands have to offer. The life expectancy of the US population across all 50 states is µ=78.62 years with a standard deviation σ=16.51 years. A randomly selected sample of 100 Hawaiians show an average life expectancy x̅=82.52 years. Is Hawaiians’ average life expectancy significantly different from 78.62, assuming α=0.01?

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Hawaiians are known to have an active lifestyle, enjoying the fresh air and natural beauty the islands have to offer. The life expectancy of the US population across all 50 states is µ=78.62 years with a standard deviation σ=16.51 years. A randomly selected sample of 100 Hawaiians show an average life expectancy x̅=82.52 years. Is Hawaiians’ average life expectancy significantly different from 78.62, assuming α=0.01?

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Step 1

State the appropriate hypothesis:

Denote the population mean life expectancy of the US population across all 50 states as μ and is 78.62 years. The standard deviation is 16.51. The sample size is 100 and the sample average life expectancy is 85.52 years.

The claim is whether or not the Hawaiians average life expectancy to be changed from 78.62.

The hypotheses are given below:

Null hypothesis:

H0 : μ = 78.62

That is, the Hawaiians average life expectancy is equal to 78.62 years.

Alternative hypothesis:

H1 : μ ≠ 78.62 (Two tail test).

That is, the Hawaiians average life expectancy is not equal to 78.62 years.

Step 2

Obtain the test statistic value.

Here, the population standard deviation is known.

Under null hypothesis the population mean life expectancy of the US population across all 50 states is μ0 = 78.62.

The sample average life expectancy is x-bar = 85.52, the sample size is n = 100.

Test statistic:

The test statistic for hypothesis test for mean is given by: help_outlineImage TranscriptioncloseZ = Tsample mean, 4 hypothesized mean a population standard deviaton, n - sample size = The test statistic value is obtained as 4.18 from the calculation given below 85.52 78.62 16.512 100 6.9 = 1.651 = 4.18 fullscreen
Step 3

Obtain the critical value:

The level of significance is α = 0.01.

The critical value is obtai...

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